Philip M. Morse
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Thermal
Physics
THERMAL
PHYSICS
A Preliminary Edition
Philip M. Morse
Professor of Physics
Massachusetts Institute of Technology
W. A. Benjamin, Inc.
New York 1962
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ENGINEERING f
THERMAL PHYSICS * PHYS ' CS
A Preliminary Edition i,BRARy ?
Copyright © 1961 by W. A. Benjamin, Inc.
All rights reserved.
Library of Congress Catalog Card Number: 6118590
Manufactured in the United States of America
The manuscript ivas received
on August 25, 1961, and published
on January 25, 1962
W. A. BENJAMIN, INC.
2465 Broadway, New York 25, New York
Preface
This book represents a stage in the process of revising the course
in heat and thermodynamics presented to physics undergraduates at
the Massachusetts Institute of Technology. It is, in its small way, a
part of the revision of undergraduate curriculum in physics which has
recently been going on at the Institute and at other centers of physics
in this country. Such major revisions must be made at least once a
generation if we are to compress the expanding field of physics into
a form that can be assimilated by the physics student by the time he
j completes his graduate education.
Such a downward compaction of new physics, introducing it earlier
in the education of a physicist, is not a task that can be undertaken
. one subject at a time. For example, the basic techniques of classical
; dynamics and electromagnetic theory could not now be taught effec
tively to juniors or seniors unless they were already fluently conver
^ sant with the differential calculus. This particular revision was ac
complished, in this country, about a generation ago; prior to that time
graduate students had to unlearn the geometric tricks they had learned
as undergraduates, before they could assimilate the methods of La
grange and Hamilton.
Now it is necessary to bring the concepts of the quantum theory
into the undergraduate curriculum, so the graduate student does not
V~) have to start over again when he takes his first graduate course in
^ atomic or nuclear physics. Again the revision must be thorough, from
the content of the freshman courses in physics and chemistry to the
^ choice of topics in electromagnetic theory and dynamics. Unless the
student becomes familiar with quantum theory and with the parts of
classical theory relevant to quantum theory while an undergraduate,
he has not the time to achieve a general understanding of modern
physics as a graduate student.
Perhaps this task of compression will eventually become impossi
ble; perhaps we shall have to give up educating physicists and resign
■ l ourselves to educating nuclear physicists or solidstate physicists or
"
i
vi PREFACE
the like. This has happened in other branches of science. It would
inevitably happen here if the task were simply that of compression of
a growing mass of disparate observed facts. But physics has so far
been fortunate enough to have the scope of its theoretical constructs
expand nearly as rapidly as the volume of its experimental data. Thus
it has so far not been necessary, for example, to have to teach two
subjects in the time previously devoted to one; it has instead been
possible to teach the concepts of a new theory which encompasses
both of the earlier separate subjects, concentrating on conceptual un
derstanding and relegating details to later specialist subjects. The
range of coverage of each course has increased; by and large the
number of concepts has not.
Of course the newer, more inclusive theories embody more
sophisticated concepts and a wider range of mathematical techniques.
So it is not an easy job to make them understandable to the undergrad
uate. However, the task is not yet impossible, in this writer's opinion.
Statistical physics is a case in point. It impinges on nearly all of
modern physics and is basic for the understanding of many aspects,
both experimental and theoretical. Classical thermodynamics is in
adequate for many of the applications; to be useful in the areas of
present active research it must be combined with quantum theory via
the concepts of statistical mechanics. And this enrichment of thermo
dynamics should be included in the undergraduate course, so that the
student can apply it in his graduate courses from the first.
Such a course needs thorough replanning, both as to choice of ma
terial and order. Topics must be omitted to make room for new items
to be added. The problem is not so much the number of different con
cepts to be taught as the abstractness and sophistication of these con
cepts. The thermodynamic part has to be compressed, of course, but
not at the price of excluding all variables except P, V, and T. Engi
neering applications have to be omitted and special cases can be rel
egated to problems. Because of its importance in statistical mechan
ics, entropy should be stressed more than might be necessary in a
course of thermodynamics alone. The kinetic theory part can be used
as an introduction to the concepts of statistical mechanics, tying the
material together with the Boltzmann equation, which recently became
important in plasma physics. In statistical mechanics the effort must
be to unify the point of view, so that each new aspect does not seem
like a totally new subject. More needs to be done than is embodied in
this text, but the class response so far has indicated that ensembles
and partition functions are not necessarily beyond the undergraduate.
As usual, the problem is best solved experimentally, by trying out
various ways of presentation to see which ones lead the student most
easily to the correct point of view— which exposition brings the sparks
of interest and understanding from the class. The problem would not
be soluble, of course, if the students do not already possess some
PREFACE vii
knowledge of atomic structure and the basic concepts of quantum me
chanics provided by a prior course in atomic structure that did not
stop at Bohr orbits.
The author has been working at this pedagogical problem, off and
on, for about five years, first taking recitation sections and, for the
past two years, giving the lectures in the springsemester senior
course in thermodynamics and statistical mechanics for physics ma
jors at the Institute. The present text is based on the lecture notes of
the past term. Obviously the presentation has not yet reached its final
form, ready for embalming in hard covers. But the results so far are
encouraging, both in regard to interest aroused in the students and to
concepts assimilated (at least long enough to use them in the final ex
amination!). It is rated as a stiff course, but in the writer's experi
ence this has never hurt a course's popularity; boredom is shunned,
not work.
The writer is indebted to the subjects of his experimentation, the
roughly 300 students who have attended his lectures during the past
two years. Their interest, their questions, and their answers on ex
amination papers have materially influenced the choice of subject
matter and its manner of presentation. He is also particularly in
debted to Professors L. Tisza and L. C. Bradley, who have read and
commented on various parts of the material, and to Mr. Larry Zamick
who has painstakingly checked over the present text. They are to be
thanked for numerous improvements and corrections; they should not
be blamed for the shortcomings that remain.
PHILIP M. MORSE
Cambridge, Massachusetts
July 1961
Contents
Preface
PART I. THERMODYNAMICS 1
1 . Introduction 3
Historical. Thermodynamics and statistical mechanics.
Equilibrium states
2. Heat, Temperature, and Pressure 7
Temperature. Heat and energy. Pressure and atomic
motion. Pressure in a perfect gas
3. State Variables and Equations of State 13
Extensive and intensive variables. Pairs of mechanical
variables. The equation of state of a gas. Other equa
tions of state. Partial derivatives
4. The First Law of Thermodynamics 23
Work and internal energy. Heat and internal energy.
Quasistatic processes. Heat capacities. Isothermal
and adiabatic processes
5. The Second Law of Thermodynamics 32
Heat engines. Carnot cycles. Statements of the second
law. The thermodynamic temperature scale
6. Entropy 40
A thermal state variable. Reversible processes. Irre
versible processes. Entropy of a perfect gas. The Joule
experiment. Entropy of a gas. Entropy of mixing
7. Simple Thermodynamic Systems 51
The JouleThomson experiment. Blackbody radiation.
Paramagnetic gas
Vlll
CONTENTS ix
8. The Thermodynamic Potentials 59
The internal energy. Enthalpy. The Helmholtz and
Gibbs functions. Procedures for calculation. Examples
and useful formulas
9. Changes of Phase 68
The solid state. Melting. ClausiusClapeyron equation.
Evaporation. Triple point and critical point
10. Chemical Reactions 78
Chemical equations. Heat evolved by the reaction. Re
actions in gases. Electrochemical processes
PART II. KINETIC THEORY 85
11. Probability and Distribution Functions 87
Probability. Binomial distribution. Random walk. The
Poisson distribution. The normal distribution
12. Velocity Distributions 96
Momentum distribution for a gas. The Maxwell distri
bution. Mean values. Collisions between gas molecules
13. The Maxwell Boltzmann Distribution 103
Phase space. The Boltzmann equation. A simple exam
ple. A more general distribution function. Mean energy
per degree of freedom. A simple crystal model. Mag
netization and Curie's law
14. Transport Phenomena 116
An appropriate collision function. Electric conductivity
in a gas. Drift velocity. Diffusion
15. Fluctuations 122
Equipartition of energy. Mean square velocity. Fluc
tuations of simple systems. Density fluctuations in a
gas. Brownian motion. Random walk. The Langevin
equation. The Fokker Planck equations
PART III. STATISTICAL MECHANICS 137
16. Ensembles and Distribution Functions 139
Distribution functions in phase space. Liouville's theo
rem. Quantum states and phase space
17. Entropy and Ensembles 147
Entropy and information. Information theory. Entropy
for equilibrium states. Application to a perfect gas
x CONTENTS
18. The Microcanonical Ensemble 154
Example of a simple crystal. Microcanonical ensemble
for a perfect gas. The Maxwell distribution
19. The Canonical Ensemble 163
Solving for the distribution function. General properties
of the canonical ensemble. The effects of quantization.
The hightemperature limit
20. Statistical Mechanics of a Crystal 170
Normal modes of crystal vibration. Quantum states for
the normal modes. Summing over the normal modes.
The Debye formulas. Comparison with experiment
21. Statistical Mechanics of a Gas 181
Factoring the partition function. The trans lational fac
tor. The indistinguishability of molecules. Counting
the system states. The classical correction factor. The
effects of molecular interaction. The Van der Waals
equation of state
22. A Gas of Diatomic Molecules 195
The rotational factor. The gas at moderate tempera
tures. The vibrational factor
23. The Grand Canonical Ensemble 202
An ensemble with variable N. The grand partition func
tion. The perfect gas once more. Density fluctuations
in a gas
24. Quantum Statistics 208
Occupation numbers. Maxwell Bo ltzmann particles.
Bosons and fermions. Comparison among the three
statistics. Distribution functions and fluctuations
25. BoseEinstein Statistics 220
General properties of a boson gas. Classical statistics
of blackbody radiation. Statistical mechanics of a pho
ton gas. Statistical properties of a photon gas. Statis
tical mechanics of a boson gas. Thermal properties of
a boson gas. The degenerate boson gas
26. FermiDirac Statistics 223
General properties of a fermion gas. The degenerate
fermion gas. Behavior at intermediate temperatures
27. Quantum Statistics for Complex Systems 238
Wave functions and statistics. Symmetric wave functions.
Antisymmetric wave functions. Wave functions and equi
librium states. Electrons in a metal. Ortho and para
hydrogen
CONTENTS xi
References 249
Problems 250
Constants 268
Glossary 270
Index 272
THERMODYNAMICS
Introduction
The subject matter of this book, thermodynamics, kinetic theory,
and statistical mechanics, constitutes the portion of physics having to
do with heat. Thus it may be called thermal physics. Since heat and
other related properties, such as pressure and temperature, are char
acteristics of aggregates of matter, the subject constitutes a part of
the physics of matter in bulk. Its theoretical models, devised to cor
respond with the observed behavior of matter in bulk (and to predict
other unobserved behavior), rely on the mathematics of probability
and statistics; thus the subject may alternately be called statistical
physics.
Historical
The part of the subject called thermodynamics has had a long and
controversial history. It began at the start of the industrial revolution,
when it became important to understand the relation between heat and
chemical transformations and the conversion of heat into mechanical
energy. At that time the atomic nature of matter was not yet under
stood and the mathematical model, developed to represent the relation
between thermal and mechanical behavior, had to be put together with
the guidance of the crude experiments that could then be made. Many
initial mistakes were made and the theory had to be drastically revised
several times. This theory, now called thermodynamics, concerns it
self solely with the macroscopic properties of aggregated matter, such
as temperature and pressure, and their interrelations, without refer
ence to the underlying atomic structure. The general pattern of these
interrelations is summarized in the laws of thermodynamics, from
which one can predict the complete thermal behavior of any substance,
given a relatively few empirical relationships, obtained by macro
scopic measurements made on the substance in question.
In the latter half of the nineteenth century, when the atomic nature
of matter began to be understood, efforts were made to learn how the
THERMODYNAMICS
macroscopic properties of matter, dealt with by thermodynamics,
could depend on the assumed behavior of constituent atoms. The first
successes of this work were concerned with gases, where the interac
tions between the atomic components were minimal. The results pro
vide a means of expressing the pressure, temperature, and other mac
roscopic properties of the gas in terms of average values of proper
ties of the molecules, such as their kinetic energy. This part of the
subject came to be called kinetic theory.
In the meantime a much more ambitious effort was begun by Gibbs
in this country, and by Boltzmann and others in Europe, to provide a
statistical correspondence between the atomic substructure of any
piece of matter and its macroscopic behavior. Gibbs called this the
ory statistical mechanics. Despite the fragmentary knowledge of
atomic physics at the time, statistical mechanics was surprisingly
successful from the first. Since then, of course, increased atomic
knowledge has enabled us to clarify its basic principles and extend its
techniques. It now provides us with a means of understanding the laws
of thermodynamics and of predicting the various relations between
thermodynamic variables, hitherto obtained empirically.
Thermodynamics and Statistical Mechanics
Thus thermodynamics and statistical mechanics are mutually com
plementary. For example, if the functional relationship between the
pressure of a gas, its temperature, and the volume it occupies is
known, and if the dependence of the heat capacity of the gas on its
temperature and pressure has been determined, then thermodynamics
can predict how the temperature and pressure are related when the
gas is isolated thermally, or how much heat it will liberate when com
pressed at constant temperature. Statistical mechanics, on the other
hand, seeks to derive the functional relation between pressure, vol
ume, and temperature, and also the behavior of the heat capacity, in
terms of the properties of the molecules that make up the gas.
In this volume we shall first take up thermodynamics, because it is
more obviously related to the gross physical properties we wish to
study. But we shall continue to refer back to the underlying micro
structure, by now well understood, to remind ourselves that the ther
modynamic variables are just another manifestation of atomic behav
ior. In fact, because it does not make use of atomic concepts, thermo
dynamics is a rather abstract subject, employing sophisticated con
cepts, which have many logical interconnections; it is not easy to un
derstand one part until one understands the whole. In such a case it is
better pedagogy to depart from strict logical presentation. Hence
several derivations and definitions will be given in steps, first pre
sented in simple form and, only after other concepts have been intro
duced, later re enunciated in final, accurate form.
INTRODUCTION 5
Part of the difficulty comes from the fact, more apparent now than
earlier, that the thermodynamic quantities such as temperature and
pressure are aggregate effects of related atomic properties. In ther
modynamics we assume, with considerable empirical justification,
that whenever a given amount of gas, in a container of given volume,
is brought to a given temperature, its pressure and other thermody
namic properties will take on specific values, no matter what has been
done to the gas previously. By this we do not mean that when the gas
is brought back to the same temperature each molecule of the gas re
turns to the same position and velocity it had previously. All we mean
is that the average effects of all the atoms return to their original val
ues, that even if a particular molecule does not return to its previous
position or velocity, its place will have been taken by another, so that
the aggregate effect is the same.
To thus assume that the state of a given collection of atoms can be
at all adequately determined by specifying the values of a small num
ber of macroscopic variables, such as temperature and pressure,
would at first seem to be an unworkable oversimplification. Even if
there were a large number of different configurations of atomic posi
tions and motions which resulted in the same measurement of tempera
ture, for example, there is no a priori reason that all, or even most,
of these same configurations would produce the same pressure. What
must happen (and what innumerable experiments show does happen) is
that a large number of these configurations do produce the same pres
sure and that thermodynamics has a method of distinguishing this sub
set of configurations from others, which do not produce the same
pressure. The distinguishing feature of the favored subset is embodied
in the concept of the equilibrium state.
Equilibrium States
A detailed specification of the position, velocity, and quantum state
of each atom in a given system is called a microstate of the system .
The definition is useful conceptually, not experimentally, for we can
not determine by observation just what microstate a system is in at
some instant, and we would not need to do so even if we could. As we
said before, many different microstates will produce the same macro
scopic effects; all we need to do is to find a method of confining our
attention to that set of microstates which exhibits the simple relations
between macroscopic variables, with which thermodynamics concerns
itself.
Consider for a moment all those microstates of a particular gas
for which the total kinetic energy of all the molecules is equal to some
value U. Some of the microstates will correspond to the gas being in
a state of turbulence, some parts of the gas having net momentum in
one direction, some in another. But a very large number of micro
6 THERMODYNAMICS
states will correspond to a fairly uniform distribution of molecular
kinetic energies and directions of motion, over all regions occupied
by the gas. In these states, which we shall call the equilibrium micro
states, we shall find that the temperature and pressure are fairly uni
form throughout the gas. It is a fact, verified by many experiments,
that if a gas is started in a microstate corresponding to turbulence, it
will sooner or later reach one of the equilibrium microstates, in which
temperature and pressure are uniform. From then on, although the
system will change from microstate to microstate as the molecules
move about and collide, it will confine itself to equilibrium micro
states. To put it in other language, although the gas may start in a
state of turbulence, if it is left alone long enough internal friction will
bring it to that state of thermodynamic quiescence we call equilibrium,
where it will remain.
Classical thermodynamics only deals with equilibrium states of a
system, each of which corresponds to a set of indistinguishable micro
states, indistinguishable because the temperature, the pressure, and
all the other applicable thermodynamic variables have the same val
ues for each microstate of the set. These equilibrium states are
reached by letting the system settle down long enough so that quanti
ties such as temperature and pressure become uniform throughout, so
that the system has a chance to forget its past history, so to speak.
Quantities, such as pressure and temperature, which return to the
same values whenever the system returns to the same equilibrium
state, are called state variables. A thermodynamic state of a given
system is thus completely defined by specifying the values of a rela
tively few state variables (which then become the independent varia
bles), whereupon the values of all the other applicable state variables
(the dependent variables) are uniquely determined. Dependent state
variable U is thus specified as a function U(x,y, ..., z) of the inde
pendent variables x, ..., z, where it is tacitly understood that the
functional relationship only holds for equilibrium states.
It should be emphasized that the fact that there are equilibrium
states, to which matter in bulk tends to approach spontaneously if left
to itself, and the fact that there are thermodynamic variables which
are uniquely specified by the equilibrium state (independent of the past
history of the system) are not conclusions deduced logically from some
philosophical first principles. They are conclusions ineluctably drawn
from more than two centuries of experiments.
Heat, Temperature,
and Pressure
In our introductory remarks we have used the words temperature
and pressure without definition, because we could be sure the reader
had encountered them earlier. Before long, however, these quantities
must be defined, by describing how they are measured and also by in
dicating how they are related to each other and to all the other quanti
ties that enter into the theoretical construct we call thermodynamics.
As mentioned earlier, this construct is so tightly knit that an adequate
definition of temperature involves other concepts and quantities, them
selves defined in terms of temperature. Thus a stepwise procedure is
required.
Temperature
The first step in a definition of temperature, for example, is to re
fer to our natural perception of heat and cold, noting that the property
seems to be one dimensional, in that we can arrange objects in a one
dimensional sequence, from coldest to hottest. Next we note that many
materials, notably gases and some liquids, perceptibly expand when
heated; so we can devise and arbitrarily calibrate a thermometer. The
usual calibration, corresponding to the centigrade scale, sets 0° at
the temperature of melting ice and 100° at the temperature of boiling
water, and makes the intermediate scale proportional to the expansion
of mercury within this range. We use such a thermometer to provide
a preliminary definition of temperature, T, until we have learned
enough thermodynamics to understand a better one.
Next we note, by experimenting on our own or by taking the word of
other experimenters, that when a body comes to equilibrium its tem
perature is uniform throughout. In fact we soon come to use uniform
ity of temperature as an experimental criterion of equilibrium.
When we turn to the question of the cause of change of temperature
the usual answer, that a rise in temperature is caused by the addition
of heat, has no scientific meaning until we define heat. And thus we
THERMODYNAMICS
arrive at the source of many of the mistakes made in the early devel
opment of thermodynamics. Heat is usually measured by measuring
the rise in temperature of the body to which the heat is added. This
sounds like a circular definition: The cause of temperature rise is
heat, which is measured by the temperature rise it causes. Actually
it is something more, for it assumes that there is something unique
called heat, which produces the same temperature change no matter
how the heat is produced. Heat can be generated by combustion, by do
ing work against friction, by sending electric current through a resis
tor, or by rapidly compressing a gas. The amount of heat produced in
each of these ways can be measured in terms of the temperature rise
of a standard body and the effects are proportional; twice the combus
tion producing twice the temperature rise, for example.
The early measurements of heat were all consistent with the theory
that heat is a "fluid" similar to the "electric fluid" which was also
being investigated at that time. The heat fluid was supposed to be
bound to some atoms in the body; it could be detached by pressure,
friction, or combustion; in its free state it would affect thermometers.
It seemed at first that the amount Q of "free heat fluid" present in a
body should be a thermodynamic state variable such as pressure or
temperature, a definite function of the independent variables that define
the equilibrium state. The Q for a particular amount of gas was sup
posed to be a specific function of the temperature and pressure of the
gas, for instance.
Later, however, it was demonstrated that heat is just one manifes
tation of the energy possessed by a body, that heat could be trans
formed into mechanical energy and vice versa. Historically, this
change in theory is reflected by the change in the units used to meas
ure heat. At first the unit was the kilogram calorie, the amount of
heat required to raise a kilogram of water from 4 to 5°C. More re
cently heat has been measured in terms of the usual units of energy,
the joule in the mks system of units. Careful measurement of the en
ergy lost to friction, or that lost in passing current through a resistor,
together with the resulting temperature rise in water placed in ther
mal contact, shows that a kilogram calorie of heat is equal to 4182
joules.
Heat and Energy
As soon as we realize that heat is just a particular manifestation
of the energy content of a body, we see that Q cannot be a state varia
ble. For we can add energy to a body in the form of heat and then take
it away in the form of mechanical energy, bringing the body back to its
initial equilibrium state at the end of such a cycle of operation. If heat
were a state variable, as much would have to be given off during the
cycle as was absorbed for Q to come back to its original value at the
HEAT, TEMPERATURE, AND PRESSURE 9
end of the cycle. But if heat can be changed to work, the net amount of
heat added during the cycle may be positive, zero, or negative, depend
ing on the net amount of work done by or on the body during the cycle.
The quantity which is conserved, and which thus is the state variable,
is U, the energy possessed by the body, which can be drawn off either
as heat or as mechanical work, depending on the circumstances. As
we shall see more clearly later, heat represents that energy content
of the body which is added or removed in disorganized form; work is
the energy added or removed in organized form; within certain limits
disorganization can be changed to organization and vice versa.
We can increase the energy of a body by elevating it, doing work
against the force of gravity. This increase is in potential energy,
which is immediately available again as work. The temperature of the
body is not changed by the elevation, so its heat content is not changed.
We can translate the potential energy into organized kinetic energy by
dropping the body; this also makes no change in its heat content. But
if the body, in its fall, hits the ground, the organized motion of the
body is changed to disorganized internal vibrations; its temperature
rises; heat has been produced. In a sense, the reason that classical
thermodynamics is usually limited to a study of equilibrium states is
because an equilibrium state is the state in which heat energy can
easily and unmistakably be distinguished from mechanical energy.
Before equilibrium is reached, sound waves or turbulence may be
present, and it is difficult to decide when such motion ceases to be
"mechanical" and becomes "thermal."
A preliminary definition of pressure can be more quickly achieved,
for pressure is a mechanical quantity; we have encountered it in hy
dromechanics and acoustics. Pressure is a simple representative of
a great number of internal stresses which can be imposed on a body,
such as tensions or torques or shears, changes in any of which repre
sent an addition or subtraction of mechanical energy to the body.
Pressure is more usually encountered in thermodynamic problems; it
is the only stress that a gas can sustain in equilibrium. The usual
units of pressure are newtons per square meter, although the atmos
phere (about 10 5 newtons per square meter) is also used.
Pressure and Atomic Motion
The pressure exerted by a gas on its container walls is a very good
example of a mechanical quantity which is the resultant of the random
motions of the gas molecules and which nonetheless is a remarkably
stable function of a relatively small number of state variables. To il
lustrate this point we shall digress for a few pages into a discussion of
kinetic theory. The pressure P on a container wall is the force ex
erted by the gas, normal to an area dA of the wall, divided by dA.
This force is caused by the collisions of the gas molecules against the
10 THERMODYNAMICS
area dA. Each collision delivers a small amount of momentum to dA;
the amount of momentum delivered per second is the force P dA.
Let us assume a very simplified model of a gas, one consisting of
N similar atoms, each of mass m and of "negligible" dimensions,
with negligible interactions between them so that the sole energy of
the ith atom is its kinetic energy of translation, (l/2)m(v x +v y + v z )
The gas is confined in a container of internal volume V, the walls of
which are perfect reflectors for incident gas atoms. By "negligible
dimensions" we mean the atoms are very small compared to the mean
distance of separation, so collisions are very rare and most of the
time each atom is in free motion. We also mean that we do not need
to consider the effects of atomic rotation. We shall call this simple
model a perfect gas of point atoms.
Next we must ask what distribution of velocities and positions of
the N atoms in volume V corresponds to a state of equilibrium. As
the atoms rebound from the walls and from each other, they cannot
lose energy, for the collisions are elastic. In collisions between the
atoms what energy one loses the other gains. The total energy of
translational motion of all the atoms,
U=m S(v 2 ix + v 2 iy + v? z ) = mSv 2 i = N<K.E.> tran (21)
is constant. The last part of this set of equations defines the average
kinetic energy <K.E.> tran of translation of an atom of the gas as be
ing U divided by the number of atoms N. (The angular brackets < >
will symbolize average values.)
As the gas settles down to equilibrium , U does not change but the
randomness of the atomic motion increases. At equilibrium the atoms
will be uniformly distributed throughout the container, with a density
(N/V) atoms per unit volume; their velocities will also be randomly
distributed, as many moving in one direction as in another. Some at
oms are going slowly and some rapidly, of course, but at equilibrium
the total x component of atomic momentum, Zmvj x , is zero; simi
larly with the total y and z components of momentum. The total x
component of the kinetic energy, (l/2)2mv? x , is not zero, however.
At equilibrium it is equal to the total y component and to the total z
component, each of which is equal to one third of the total kinetic en
ergy, according to Eq. (21),
V \ m V ix = \ N< K ' E ' >tr an (2 " 2)
At equilibrium all directions of motion are equally likely.
HEAT, TEMPERATURE, AND PRESSURE
11
Pressure in a Perfect Gas
Next we ask how many atoms strike the area <±A of container wall
per second. For simplicity we orient the axes so that the positive x
axis points normally into dA (Fig. 21). Consider first all those atoms
> z
FIG. 21. Reflection of atoms, with velocity v, from area
element dA in the yz plane.
in V which have their x component of velocity, vix, equal to some
value v x (v x must be positive if the atom is to hit dA). All these kinds
of atoms, which are a distance v x from the wall or closer, will hit the
wall in the next second, and a fraction proportional to dA of those will
hit dA in the next second. In fact a fraction (v x /V) dA of all the
atoms in V which have x component of velocity equal to v x will hit
dA per second. Each of these atoms, as it strikes dA during the sec
ond, rebounds with an x component v x , so each of these atoms im
parts a momentum 2mv x to dA. Thus the momentum imparted per
second by the atoms with x velocity equal to v x is 2mv x (v x / V) dA
times the total number of atoms in V having x velocity equal to v x .
And therefore the total momentum imparted to dA per second is the
sum of (2mv? / V) dA for each atom in V that has a positive value
of Vix.
Since half the atoms have a positive value of v^, the total momen
tum imparted to dA per second is
12 THERMODYNAMICS
N
2 jp! (2mv x /V) dA =  (N/V) < K.E . > tran dA =  (U/ V) dA
(23)
where we have used Eq. (22) to express the result in terms of the
mean atomic kinetic energy, defined in Eq. (21). Since this is equal
to the force P dA on dA, we finally arrive at an equation relating the
pressure P of a perfect gas of point atoms in a volume V, in terms
of the mean kinetic energy <K.E.>f ran of translation per atom or in
terms of the total energy content of the gas per unit volume (U/V)
(total as long as the only energy is kinetic energy of translation, that
is):
P=(U/V) or PV= u=N<K.E.> tran (24)
This is a very interesting result, for it demonstrates the great
stability of the relationships between aggregate quantities such as P
and U for systems in equilibrium. The relationship of Eq. (24) holds
no matter what the distribution in speed the atoms have as long as
their total energy is U, as long as the atoms are uniformly distrib
uted in space, and as long as all directions of motion are equally likely
(i.e., as long as the gas is in equilibrium). Subject to these provisos,
every atom could have kinetic energy <K.E.>tran> or na ^ °* them
could have kinetic energy (l/2)<K.E.>t ran and the other half energy
(3/2)<K.E.>t ran , or any other distribution having an average value
<K.E.> tran . As long as it is uniform in space and isotropic in direc
tion the relation between P, V, and N<K.E.>t ran is that given in Eq.
(24). Even the proportionality constant is fixed; PV is not just pro
portional to N<K.E.>j ran — the factor is 2/3, no matter what the ve
locity distribution is.
From our earlier discussion we may suspect that <K.E.>t ran is a
function of the gas temperature T; if T is increased, the kinetic en
ergy of the gas atoms should increase. We shall see what this relation
ship is in the next chapter.
State Variables and
Equations of State
To recapitulate, when a thermodynamic system is in a certain equi
librium state, certain aggregate properties of the system, such as
pressure and temperature, called state variables, have specific values,
determined only by the state and not by the previous history of the sys
tem. Alternately, specifying the values of a certain number of state
variables specifies the state of the system; the number of variables,
required to specify the state uniquely, depends on the system and on
its constraints. For example, if the system is a definite number of
like molecules in gaseous form within a container, then only two vari
ables are needed to specify the state— either the pressure of the gas
and the volume of the container, or the pressure and temperature of
the gas, or else the volume and temperature. If the system is a mix
ture of two gases (such as hydrogen and oxygen) which react chemi
cally to form a third product (such as water vapor), the relative abun
dance of two of the three possible molecular types must be specified,
in addition to the total pressure and volume (or P and T, or T and
V), to determine the state. If the gas is paramagnetic, and we wish to
investigate its therm omagnetic properties, then the strength 3C of the
applied magnetic field (or else the magnetic polarization of the gas)
must also be specified.
Extensive and Intensive Variables
One state variable is simply the amount of each chemical substance
present in the system. The convenient unit for this variable is the
mole; 1 mole of a substance, which has molecular weight M, is M
kilograms of the substance (1 mole of hydrogen gas is 2 kg of H 2 , 1
mole of oxygen is 32 kg of 2 ). By definition, each mole contains the
same number of molecules, N = 6 x 10 26 , called Avogadro's number.
In many respects a mole of gas behaves the same, no matter what its
composition. When the thermodynamic system is made up of a single
substance then the number of moles present (which we shall denote by
13
14
THERMODYNAMICS
the letter n) is constant. But if the system is a chemically reacting
mixture the n's for the different substances may change.
State variables are of two sorts, one sort directly proportional to
n, the other not. For example, suppose we have two equal amounts of
the same kind of gas, each of n moles and each in equilibrium at the
same temperature T in containers of equal volume V. We then con
nect the containers so the two samples of gas can mix. The combined
system now has 2n moles of gas in a volume 2V, and the total inter
nal energy of the system is twice the internal energy U of each origi
nal part. But the common temperature T and pressure P of the mixed
gas have the same values they had in the original separated states.
Variables of the former type, proportional to n (such as U and V),
are called extensive variables ; those of the latter type (such as T and
P) are called intensive variables. At thermodynamic equilibrium the
intensive variables have uniform values throughout the system.
A basic state variable for all thermodynamic systems (almost by
definition) is its temperature T, which is an intensive variable. At
present we have agreed to measure its value by a thermometer; a
better definition will be given later. Related to the temperature is the
heat capacity of the system, the heat required to raise the system 1
degree in temperature. Because heat is not a state variable, the
amount of heat required depends on the way the heat is added. For
example, the amount of heat required to raise T by 1 degree, when
the volume occupied by the system is kept constant, is called the heat
(T/0)
FIG. 31. Heat capacity at constant volume C v of a solid, in
units of 3nR, where R is the gas constant [see Eq.
(31)] as a function of temperature, in units of the
characteristic temperature of the solid (see Fig.
201).
STATE VARIABLES AND EQUATIONS OF STATE 15
capacity at constant volume and is denoted by C v . The heat required
to raise T 1 degree when the pressure is kept constant is called the
heat capacity at constant pressure and is denoted C p . A system at
constant pressure usually expands when heated, thus doing work, so
Cp is usually greater than C v .
These heat capacities are state variables, in fact they are extensive
variables; their units are joules per degree. The capacities per mole
of substance, c v = (C v /n) and c p = (Cp/n), are called specific heats,
at constant volume or pressure, respectively. They have been meas
ured, for many materials, and a number of interesting regularities
have emerged. For example, the specific heat at constant volume, c v ,
for any monatomic gas is roughly equal to 12,000 joules per degree
centigrade per mole, independent of T and P over a wide range of
states, whereas c v for diatomic gases is roughly 20,000 joules per
degree centigrade per mole, with a few exceptions. A typical plot of
c v for a solid is given in Fig. 31, showing that c v is independent of
T for solids only when T is quite large (there are more exceptions to
this rule with solids than with gases).
Pairs of Mechanical Variables
Other state variables are of a mechanical, rather than thermal,
type. For example, there is the pressure P (in newtons per square
meter), an intensive variable appropriate for fluids, although applica
ble also for solids that are uniformly compressed (in general, in sol
ids, one needs a tensor to describe the stress). Related to P is the
extensive variable V (in cubic meters), the volume occupied by the
system. The pair define a mechanical energy; work P dV (in joules)
is done by the system on the container walls if its volume is increased
by dV when it is in equilibrium at pressure P. The pair P and V are
the most familiar of the mechanical state variables. For a bar of ma
terial, the change in dimensions may be simple stretching, in which
case the extensive variable would be the length L of the bar, the in
tensive variable would be the tension J, and the work done on the bar
by stretching it an additional amount dL would be J dL.
Or possibly the material may be polarized by a magnetic field. The
intensive variable here is the impressed magnetic intensity 3C (in am
pere turns per meter) and the extensive variable is the total magneti
zation an of the material. Reference to a text on electromagnetic the
ory will remind us that, related to 3C, there is the magnetic induction
field ca> (in webers per square meter). For paramagnetic material of
magnetic susceptibility x» a magnetic field causes a polarization (P of
the material, which is related to .fC and (B through the equation
(B = /x 3C(1 + x) = Mo( ,,JC+ (p ), where n is the permeability of vacuum
47T x 10~ 7 henrys per meter. The total energy contributed to the mate
rial occupying volume V, by application of field x, is (1/2) jlx V 0C(P,
16 THERMODYNAMICS
exclusive of the "energy of the vacuum" (1/2) /j. V3C 2 , the total mag
netic energy being (1/2)3C(BV.
Suppose we define the total magnetization of the body as being the
quantity an = ii V (P (in webermeters); for paramagnetic materials an
would equal jll V~x3C. Then the magnetic work done on the body in mag
netic field 3C, when its magnetization is increased by d 9TC, would be
JC dan, the integral of which, for an = jli Vx3C, becomes (1/2) jlx V3C(P, as
desired. Magnetization on is thus the extensive variable related to oc.
A similar pair of state variables can be devised for dielectrics and the
electric field.
There is also an intensive variable, related to the variable n, the
number of moles of material in the system. If we add dn moles of
material to the system we add energy \i dn, where ju. is called the
chemical potential of the material. Its value can be measured by meas
uring the heat generated by a chemical reaction, as will be shown in
Chapter 10.
As we mentioned earlier, we need to determine experimentally a
certain minimum number of relationships between the state variables
of a system before the theoretical machinery of thermodynamics can
"take hold" to predict the system's other thermal properties. One of
these relationships is the dependence of one of the heat capacities,
either Cy or Cp (or Cl or Cj, or C^ or Can depending on the me
chanical or electromagnetic variables of interest) on T and on P or V
(or L or J, or on 3C or 911, as the case may be). We shall show later
that, if C v is measured, Cp can be computed (and similarly for the
other pairs of heat capacities); thus only one heat capacity needs to be
measured as a function of the independent variables of the system.
The Equation of State of a Gas
Another necessary empirical relationship is the relation between
the pair of mechanical variables P and V (or J and L, or 3C and 9Tl)
and the temperature T for the system under study. Such a relation
ship, expressed as a mathematical equation, is called an equation of
state. There must be an equation of state known for each pair of me
chanical variables of interest. We shall write down some of those of
general interest, which will be used often later.
Parenthetically it should be noted that although these relationships
are usually experimentally determined, in principle it should be pos
sible to compute them from our present knowledge of atomic behavior,
using statistical mechanics.
The equation of state first to be experimentally determined is the
one relating P, V, and T for a gas, discovered by Boyle and by
Charles. The relation is expressed by the equation PV = nR(T + T ),
where, if T is in degrees centigrade, constant R is roughly 8300
joules per degree centigrade per mole and T is roughly 273 C. This
STATE VARIABLES AND EQUATIONS OF STATE 17
suggests that we change the origin of our temperature scale from 0°C
to 273°C. The temperature measured from this new origin (called
absolute zero) is called absolute temperature and is expressed in de
grees Kelvin. For T in degrees Kelvin this equation of state is
PV = nRT (31)
Actually this is only a rough approximation of the equation of state
of an actual gas. Experimentally it is found to be a pretty good ap
proximation for monatomic gases, like helium. Moreover, the ratio
PV/nT for any gas approaches the value 8315 joules per degree Kel
vin per mole as P/T approaches zero. Since a gas is ''most gassy"
when the pressure is small and the atoms are far apart, we call Eq.
(31) the perfect gas law and call any gas that obeys it a perfect gas.
We could, alternately, use Eq. (31) as another (but not the final) defi
nition of temperature; T is equal to PV/nR for a perfect gas.
We now are in a position to illustrate how kinetic theory can supple
ment an empirical thermodynamic formula with a physical model. In
the previous chapter we calculated the pressure exerted by a gas of N
point particles confined at equilibrium in a container of volume V.
This should be a good model of a perfect gas. As a matter of fact Eq.
(24) has a form remarkably like that of Eq. (31). All we need to do
to make the equations identical is to set
nRT = u=N<K.E.> tran
The juxtaposition is most suggestive. We have already pointed out that
N, the number of molecules, is equal to nN , where N , Avogadro's
number, is equal to 6 x 10 26 for any substance. Thus we reach the re
markably simple result, that RT = (2/3)N <K.E.>t r an for an Y perfect
gas. For this model, therefore, the average kinetic energy per mole
cule is proportional to the temperature, and the proportionality con
stant (3/2)(R/N ) is independent of the molecular mass. The ratio
(R/N ) = 1.4 x 10" 23 joules per degree Kelvin is called k, the Boltz
mann constant.
Thus the model suggests that for those gases which obey the perfect
gas law fairly accurately, the average kinetic energy of molecular
translation is directly proportional to the temperature,
<K.E.> tran = kT perfect gas (32)
independent of the molecular mass. Only the kinetic energy of trans
lation enters into this formula; our model of point atoms assumed their
rotational kinetic energy was negligible. We might expect that this
18 THERMODYNAMICS
would be true for actual monatomic gases, like helium and argon, and
that for these gases the total internal energy is
U = N<K.E.> tran = NkT = nRT (33)
Measurement shows this to be nearly correct [see discussion of Eq.
(611)] . For polyatomic gases U is greater, corresponding to the ad
ditional kinetic energy of rotation (the additional term does not enter
into the equation for P, however). We shall return to this point, to
enlarge on and to modify it, as we learn more.
Other Equations of State
Of course the equation of state for an actual gas is not as simple as
Eq. (31). We could, instead of transforming our measurements into
an equation, simply present the relationship between P, V, and T in
the form of a table of numbers or a set of curves. But, as we shall
see, the thermal behavior of bodies usually is expressed in terms of
the first and second derivatives of the equation of state, and taking de
rivatives of a table of numbers is tedious and subject to error. It is
often better to fit an analytic formula to the data, so we can differen
tiate it more easily.
A formula that fits the empirical behavior of many gases, over a
wider range of T and P than does Eq. (31), is the Van der Waals
approximation,
(V  nb)(p + ~) = nRT (34)
For large enough values of V this approaches the perfect gas law.
Typical curves for P against V for different values of T are shown
in Fig. 32. For temperatures smaller than (8a/27bR) there is a range
of P and V for which a given value of pressure corresponds to three
different values of V. This represents (as we shall see later) the
transition from gas to liquid. Thus the Van der Waals formula covers
approximately both the gaseous and liquid phases, although the accu
racy of the formula for the liquid phase is not very good.
It is also possible to express the equation of state as a power series
in (1/V),
(nRT/ V)
n _/_x . n
l + ^B(T)+ li] C(T) + 
This form is called the virial equation and the functions B(T), C(T),
etc., are called virial coefficients. Values of these coefficients and
STATE VARIABLES AND EQUATIONS OF STATE 19
0.2
0.1
1
1 1 '
~~
1 \ \ t  (8/27)
V <= ^
\ r n 25
\ < ^^' 00 ^ m Q ?, , 
~ ™ ' ■
\</^ _ J ^""' — ^~ — ■ " —
1
1 / /l 1 1 1 1 1 1 1 1
(V/nb)
10
FIG. 32. The Van der Waals equation of state. Plots of b 2 P/a
versus V/nb for different values of t = RbT/a.
Point C is the critical point (see Fig. 93).
their derivatives can then be tabulated or plotted for the substance
under study.
Corresponding equations of state can be devised for solids. A sim
ple one, which is satisfactory for temperatures and pressures that are
not too large (below the melting point for T, up to several hundred at
mospheres for P), is
V = V (l +j3T  kP)
(36)
Both 0, which is called the thermal expansion coefficient , and k,
called the compressibility, are small quantities, of the order of 10" 6
for metals, for example. They are not quite constant; they vary as T
and P are changed, although the variation for most solids is not large
for the usual range of T and P.
The other pairs of mechanical variables also have their equations
of state. For example, in a stretched rod the relation between tension
20 THERMODYNAMICS
J and length L and temperature T is, for stretches within the elastic
limit,
J = (A + BT)(L L ) (37)
where A, B, and L are constants (approximately); B is negative for
many substances but positive for a few, such as rubber.
Likewise there is a magnetic equation of state, relating magnetic
intensity 3C, magnetization 3TC, and T. For paramagnetic materials,
for example, Curie's law,
m = (nD.TC/T) (38)
is a fairly good approximation, within certain limits for T and 3C. The
Curie constant D is proportional to the magnetic susceptibility of the
substance.
Partial Derivatives
In all these equations there is a relationship between at least three
variables. We can choose any pair of them to be the independent vari
ables; the other one is then a dependent variable. We shall often wish
to compute the rate of change of a dependent variable with respect to
one of the independent variables, holding the other constant. This rate,
called a partial derivative, is discussed at length in courses in ad
vanced calculus. In thermodynamics, since we are all the time chang
ing from one pair of independent variables to another, we find it advis
able to label each partial by both independent variables, the one varied
and the one held constant. The partial (3P/8V)rp, for example, is the
rate of change of P with respect to V, when T is held constant; V
and T are the independent variables in this case, and P is expressed
explicitly as a function of V and T before performing the differenti
ation.
There are a number of relationships between partial derivatives
that we shall find useful. If z and u are dependent variables, func
tions of x and y, then, by manipulation of the basic equation
dz = (3z/3x) y dx + (3z/8y) x dy
we can obtain
(39)
The last equation can be interpreted as follows: On the left we express
x as a function of y and z, on the right z is expressed as a function
3z\
(du/ax)y _
(8u/3z) y
1
() =
\ey/z
(3z/3y)x
(ax/az) y '
(dz/9x)y
STATE VARIABLES AND EQUATIONS OF STATE 21
of x and y before differentiating and the ratio is then reconverted to
be a function of y and z to effect the equation. Each partial is itself
a function of the independent variables and thus may also be differen
tiated. Since the order of differentiation is immaterial, we have the
useful relationship
L /iz\ "I . r_a_/Bz\
(310)
As an example of the use of these formulas, we can find the partial
(3V/3T)p, as function of P and V or of T and V, for the Van der
Waals formula (34):
\BTJ }
(9P/9T) V nR/(V  nb)
(3P/3V) T [nRT/(V  nb) 2 ]  (2an 2 /V 3 )
R(V  nb)V 3
RTV 3  2an(V  nb)<
We shall often be given the relevant partial derivatives of a state
function and be required to compute the function itself by integration.
If (3z/3x)y = f(x) and (3z/8y) x = g(y) this is straightforward; we inte
grate each partial separately and add
z = ft (x) dx + / g(y) dy
But if either partial depends on the other independent variable it is
not quite so simple. For example, if (3z/8x) y = f(x) + ay and
(8z/3y) x = ax, then the integral is
z = Jf(x) dx + axy [not Jf(x) dx + 2axy]
as may be seen by taking partials of z. The cross term appears in
both partials and we include it only once in the integral. To thus co
alesce two terms of the integral, the two terms must of course be
equal. This seems to be assuming more of a relationship between
(az/3x)y and (az/3y) x than we have any right to do, until we remem
ber that they are related, according to Eq. (310), in just the right way
so the cross terms can be coalesced. When this is so, the differential
dz = (az/3x) y dx + (3z/8y) x dy is a perfect differential, which can be
integrated in the manner just illustrated to obtain z, a function of x
and y, the integrated value coming out the same no matter what path
in the x,y plane we choose to perform the integration along, as long as
the terminal points of the path are unchanged (Fig. 33).
A differential dz = f(x,y) dx + g(x,y) dy, where (3f/8y) x is not equal
to (3g/3x)y, results in an integral which depends on the path of integra
22
THERMODYNAMICS
tion as well as the end points, is called an imperfect differential, and is
distinguished by the bar through the d. The integral of the perfect dif
ferential dz = y dx + x dy, from (0,0) to (a,b) over the path from (0,0)
(a,0)
FIG. 33. Integration in the xy plane.
b a
to (0,b) to (a,b) is • / dy + b • J dx = ab, which equals that for the path
a b
(0,0), (a,0), (a,b), 0/ dx + a/dy = ab. On the other hand, the integral
from (0,0) to (a,b) of the imperfect integral d"z = y dx  x dy is ab
over the first route and ab over the second. Such a differential can
not be integrated to produce a state function z(x,y). However, we can
multiply the imperfect differential d"z by an appropriate function of x
and y (in this case 1/y 2 ), which will turn it into a perfect differential,
du; in this case
(CTz/y 2 ) = du = (1/y) dx  (x/y 2 ) dy
and
u = (x/y)
The factor that converts an imperfect differential into a perfect one is
called an integrating factor . One always exists (although it may be
hard to find) for differentials of two independent variables. For more
than two independent variables there are imperfect differentials for
which no integrating factor exists.
The First Law of
Thermodynamics
An important state function is the internal energy U of the system .
For a perfect gas of point atoms, Eq. (33) indicates that U = (3/2)nRT,
if T and V, or T and P, are the independent var iables ; U= (3/2)PV
if P and V are. The internal energy of a system is an extensive vari
able.
Work and Internal Energy
The internal energy U can be changed by having the system do
work dW against some externally applied force, or by having this
force do work dW on the system. For example, if the system is con
fined under uniform pressure, an increase in volume would mean that
the system did work £IW = P dV; if the system is under tension J, it
would require work dW = J dL to be done on the system to increase
its length dL. Similarly an increase in magnetization dan in the pres
ence of a field 3C will increase U by jc dan. Or, if dn moles of a sub
stance with chemical potential /i is added, U would increase by /i dn.
In all these cases work dW is being done in an organized way by the
system and U is increased by dW. Note our convention, a positive
dW is work done by the system, a negative value represents work
done on the system, so that the change in U is opposite in sign toflW.
Note also that we have been using the symbol of the imperfect dif
ferential for AW, implying that the amount of work done by the system
depends on the path (i.e., on how it is done). For example, the work
done by a perfect gas in going from state 1 of Fig. 4 1 to state 2 dif
fers whether we go via path a or path b. Along path la, V does not
change, so no work is done by or on the gas, although the temperature
changes from T x = (P 1 V 1 /nR) to T a = (P 2 V 1 /nR). Along path a^,, P
does not change, so that the work done by the gas in going along the
whole of path la2 is AW a = P 2 (V 2  V x ). Similarly, the work done by
the gas in going along path lb2 is AW^ = P X (V 2  V x ), differing from
AW a by the factor P x . This same sort of argument can be used to
23
24 THERMODYNAMICS
show that work done by the system, in consequence of a variation of
any of the mechanical variables that describe its state, cannot be a
state variable.
Something more should be said about the meaning of the diagram
of Fig. 41. In our calculations we tacitly assumed that at each point
along each path the system was in an equilibrium state, for which the
equation of state PV = nRT held. But for a system to be in equilib
rium, so that P and T have any meaning, it must be allowed to settle
down, so that P and V (and therefore T) are assumed constant. How,
then, can we talk about going along a path, about changing P and V,
and at the same time assume that the system successively occupies
equilibrium states as the change is made? Certainly if the change is
made rapidly, sound waves and turbulence will be created and the
equation of state will no longer hold. What we must do is to make the
change slowly and in small steps, going from 1 to i and waiting till
the system settles down, then going slowly to j, and so on. Only in
the limit of many steps and slow change can we be sure that the sys
tem is never far from equilibrium and that the actual work done will
approach the value computed from the equation of state. In thermody
namics we have to limit our calculations to such slow, stepwise
changes (called quasistatic processes) in order to have our formulas
hold during the change. This may seem to be an intolerable limitation
on the kinds of processes thermodynamics can deal with; we shall see
later that the limitation is not as severe as it may seem.
Heat and Internal Energy
If the only way to change the system's energy is to perform work
on it or have it do work, then the picture would be simple. Not only dU
but also dW would be a perfect differential; whatever work was per
formed on the system could eventually be recovered as mechanical (or
electrical or magnetic) energy. This was the original theory of ther
modynamic systems; work was work and heat was heat. The introduc
tion of heat dQ served to raise the temperature of the body (indeed
the rise in temperature was the usual way in which the heat added
could be measured, as was pointed out at the beginning of Chapter 2),
and when the body was brought back to its initial temperature it would
have given up the same heat that had been given it earlier. We could
thus talk about an internal energy of the system, which was the net
balance of work done on or by the system, and we could talk about the
heat possessed by the body, the net balance of heat intake and output,
measured by the body's temperature.
It was quite a shock to find that this model of matter in bulk was
inconsistent with observation. A body's temperature could be changed
by doing work on it; a body could take in heat (from a furnace, say)
and produce mechanical work. It was realized that we cannot talk
THE FIRST LAW OF THERMODYNAMICS 25
1
1
1 >w
b
1
1 >
i
J
v
V
a
1
2
FIG. 41. Plots of quasistatic processes on the PV plane.
about the heat "contained" by the system, nor about the mechanical
energy it contains. It possesses just one pool of contained energy,
which we call its internal energy U, contributed to by input of both
mechanical work and also of heat, which can be withdrawn either as
mechanical energy or as heat. Any change in U, dU is the difference
between the heat added, d"Q, and the work done by the system tlW dur
ing a quasistatic process,
dU = dQ  aw
= dQ  P dV + J dL + H dM + /i dn +
(41)
where dU is a perfect differential and dQ and TXW are imperfect
ones. Note the convention used here; dQ is the heat added to the sys
tem, "dW is the work done by the system.
This set of equations is the first law of thermodynamics . It states
that mechanical work and heat are two forms of energy and must be
lumped together when we compute the change in internal energy of the
system. It was not obvious to physicists of the early nineteenth cen
tury. To have experiments show that heat could be changed into work
ad libitum, that neither dQ nor dW were perfect differentials, seemed
at the time to introduce confusion into a previously simple, symmetric
theory.
There were some compensations. Gone was the troublesome ques
tion of how to measure the heat "contained" by the system. The ques
tion has no meaning; there is no state variable Q, there is only inter
nal energy U to measure. Also the amount of heat dQ added could
26 THERMODYNAMICS
sometimes be most easily measured by measuring dU and 3W and
computing dQ = dU + dW. An accurate measurement of the amount of
heat added is even now difficult to make in many cases (the heat pro
duced by passage of electric current through a resistance is relatively
easy to measure, but the direct measurement of heat produced in a
chemical reaction is still not easy).
Of course, compensations or not, Eq. (41) was the one that corre
sponded with experiment, so it was the one to use, and people had to
persuade themselves that the new theory was really more "simple"
and * 'obvious " than the old one. By now this revision of simplicity has
been achieved; the idea of heat as a separate substance appears to us
"illogical."
Just as with work, the total amount of heat added or withdrawn from
a system depends on the process, on the path in the P,V plane of Fig.
41, for example. Of course the process must be that slow, stepwise
kind, called quasistatic, if we are to use our thermodynamic formulas
to calculate its change. To go from 1 to a in Fig. 41 we must remove
enough heat from the gas, keeping its volume constant meanwhile, to
lower its temperature from T x = (P^/nR) to T a = (P 2 V 1 /nR). We
could do this relatively quickly (but not quasistatically) by placing the
gas in thermal contact with a constant temperature heat source at
temperature T a . Such a source, sometimes called a heat reservoir,
is supposed to have such a large heat capacity that the amount of heat
contributed by the gas will not change its temperature. In this case
the gas would not be in thermal equilibrium until it settled down once
more into equilibrium at T = T a . To carry out a quasistatic process,
for which we could use our formulas to compute the heat added, we
should have to place the gas first into contact with a heat reservoir at
temperature T x  dT, allowing it to come to equilibrium, then place
it in contact with a reservoir at T x  2dT, and so on.
To be sure, if the gas is a perfect gas of point atoms, we already
know that U = (3/2)PV, so that U a  U x = (3/2)V 1 (P 1  P 2 ), whether
the system passes through intermediate equilibrium states or not, as
long as states 1 and a are equilibrium states. Then since in this case
fiW = 0, we can immediately find dQ. But if we did not know the for
mula for U, but only knew the heat capacity of the gas at constant vol
ume, we should be required (conceptually) to limit the process of going
from 1 to a to a quasistatic one, in order to use C v to compute the
heat added. For a quasistatic process, for a perfect gas of point
atoms where C v = (3/2)nR, the heat added to the gas between 1 and a is
Qla = /c v dT = (nR) / (V x dP/nR) =  V 1 (P 1  P 2 )
checking with the value calculated from the change in U.
THE FIRST LAW OF THERMODYNAMICS 27
Quasistatic Processes
In going from a to 2 the same problem arises. We can imagine
that the gas container is provided with a piston, which can be moved
to change the volume V occupied by the gas. We could place the gas in
thermal contact with a heat reservoir at temperature T 2 = (P 2 V 2 /nR)
and also move the piston so the volume changes rapidly from Vj to V 2
and then wait until the gas settles down to equilibrium. In this case we
can be sure that the internal energy U will end up having the value
(3/2)nRT 2 = (3/2)P 2 V 2 , but we cannot use thermodynamics to compute
how much work was done during the process or how much heat was
absorbed from the heat reservoir. If, for example, instead of moving
a piston, we turned a stopcock and let the gas expand freely into a
previously evacuated volume (V 2  Vj, the gas would do no work while
expanding. Whereas, if we moved the piston very slowly, useful work
would be done and more heat would have to be taken from the reser
voir in order to end up with the same value of U at state 2. In the
case of free expansion, the energy not given up as useful work would
go into turbulence and sound energy, which would then degenerate into
heat and less would be taken from the reservoir by the time the sys
tem settled down to state 2.
If we did not know how U depends on P and T, but only knew the
value of the heat capacity at constant pressure [which we shall show
later equals (5/2)nR for a perfect gas of point atoms] we should have
to devise a quasistatic process, going from a to 2, for which to com
pute AQ a 2 and AW a2 and thence, by Eq. (41), to obtain AU. For ex
ample, we can attach the piston to a device (such as a spring) which
will maintain a constant pressure P 2 on the gas no matter what posi
tion the piston takes up (such a device could be called a constant
pressure work source, or a. work reservoir). We then place the gas in
contact with a heat reservoir at temperature T a + dT, wait until the
gas comes to equilibrium at slightly greater volume, place it in con
tact with another reservoir at temperature T a + 2dT, and so on. The
work done in this quasistatic process at constant pressure is, as we
said earlier, AW a 2 = P 2 (V 2  Vj). The heat donated by the heat reser
voir [if C p = (5/2)nR] is AQ a2 = (5/2)nR / dT= (5/2)P 2 (V 2  V x ) and
the difference is AU= AQ a2  AW a2 = (3/2)P 2 (V 2  Vj, as it must be.
Thus thermodynamic computations, using an appropriate quasi
static process, can predict the change in internal energy U (or in any
other state variable) for any process, fast or slow, which begins and
ends in an equilibrium state. But these calculations cannot predict the
amount of intake of heat or the production of work during the process
unless the process differs only slightly from the quasistatic one used
in the calculations. It behooves us to avoid incomplete differentials,
such as dW and dQ, and to express the thermodynamic changes in a
in a system during a process in terms of state variables, which can
28 THERMODYNAMICS
be computed for any equilibrium state, no matter how the system ac
tually arrived at the state.
Heat Capacities
To integrate U for a simple system, where
dU = dQPdV (42)
we need to work out some relationships between the heat capacities
and the partial derivatives of U. For example, if T and V are chosen
to be the independent variables, the heat absorbed in a quasistatic
process is
dQ = dU + P dV = (8U/8T) V dT + [(aU/aV) T + P] dV (43)
Since C v is defined as the heat absorbed per unit increase in T,
when dV = 0, we see that
C V =(3U/3T) V (44)
so that Eq. (43) can be written
8Q = C v dT + [(3U/9V) T + P] dV (45)
If T and V are varied so that P remains constant, then when T
changes by dT, V will change by (dV/dT)* dT and the amount of heat
absorbed is
dQ = C p dT = C v dT + [(3U/aV) T + P] (aV/3T) p dT or
or
C p = C v + (aV/3T) p [(3U/aV) T + P] (46)
In our earlier discussion we stated that for a perfect gas of point
atoms C v = (3/2)nR and C p  (5/2)nR; we can now justify our state
ments. From Eq. (33) we know that for such a gas U = (3/2)nRT, so
Eq. (44) gives us C v immediately. It also shows that, for this gas
(3U/3V) T = 0, so that, from Eq. (46),
C v = nR; C p = C v + (^)y = nR (47)
for a perfect gas of point atoms.
A similar set of relationships can be derived for other pairs of
mechanical variables. For example, for paramagnetic materials, the
specific heats for constant SHI and for constant 3C are obtained from
Eq. (41) (assuming that V, L, and n are constant):
THE FIRST LAW OF THERMODYNAMICS
29
tfQ = (aU/aTjc^dT + [(3U/89TC) T  3C] d9TC
from which we can obtain
911
V9TV
3C
911
5fC
L
(48)
For a material obeying Curie's law 9TC = (nDac/T), it again turns out
that (SU/39il)x = 0> analogous to the perfect gas, so that
3C
C w + (nD3C 2 /T 2 ) = C cw ,+ (l/nD)9U 2
9H
(49)
but since we are
Strictly speaking, Cg^ should be written Cyc^L
usually concerned with one pair of variables at a time, no ambiguity
arises if we omit all but the variables of immediate interest.
Isothermal and Adiabatic Processes
Other quasistatic processes can be devised beside those at constant
volume and at constant pressure. For example, the system may be
placed in thermal contact with a heat reservoir and the mechanical
variables may be varied slowly enough so that the temperature of the
system remains constant during the process. This is called an iso
thermal process. A heat capacity for this process does not exist (for
mally speaking, C^ is infinite). However it is important to be able to
calculate the relationship between the heat dQ absorbed from the res
ervoir and the work tlW done by the system while it proceeds.
For the perfect gas, where (3U/3V)x  0, and for paramagnetic
materials, where (3U/39tc)t = 0, and for other systems where U turns
out to be a function of T alone, the heat absorbed from the reservoir
during the isothermal process exactly equals the work done by the sys
tem. Such systems are perfect isothermal energy transformers,
changing work into heat or vice versa without holding out any of it
along the way. The transformation cannot continue indefinitely, how
ever, for physical limits of volume or elastic breakdown or magnetic
saturation or the like will intervene.
For less simple substances the heat absorbed in an element of an
isothermal process is
dQ = dU + flW =
.3 V /Tarn:
+ P
dv +
3U\
.3 911/ rpy
 H
dan
+ E
i
©TV...'' 1  1 .
dn A +
(410)
differing from the work done by the amount by which U increases as
30 THERMODYNAMICS
V or M or nj is changed isothermally. We remind ourselves that
jLLidn^ is the chemical energy introduced into the system when dni
moles of substance i is introduced or created in the system, and thus
that iii dn^ is the chemical analogue of work done.
Another quasistatic process can be carried out with the system iso
lated thermally, so that dQ is zero. This is called an adiabatic proc
ess; for it the heat capacity of the system is zero. The relationship
between the variables can be obtained from Eq. (41) by setting
d"Q = 0. For example, for a system with V and T as independent va
riables, using Eqs. (45) and (46), the change of T with V in an
adiabatic process is
<=?«&i^"  (8). ci )(H) P
(411)
where y  (C p /C v ) is a state variable. The reason for using the sub
script s to denote an adiabatic process will be elucidated in Chapter 6
We see that when y is constant (as it is for a perfect gas) the adia
batic change of T with V is proportional to the change of T with V
at constant pressure.
For the perfect gas of point atoms, where C p = (5/2)nR and
C v = (3/2)nR, y = 5/3 and (aT/3V) p = (P/nR) = (T/V), the relation be
tween T and V for an adiabatic expansion is
(dT/T) + (y  l)(dV/V) = or TV r " 1 = (PV>7nR) = const.
(412)
Compressing a gas adiabatically increases its temperature, because
y > 1 pressure increases more rapidly, with change of volume, in an
adiabatic compression than in an isothermal compression, where
(PV/nR) is constant.
Similarly, for a paramagnetic material that obeys Curie's law and
happens to have Cg^ independent of T and am, the relation between T
and 9TC during adiabatic magnetization is [see Eqs. (48) and (49)]
C gTl dT=Hd3TC or C^dT = — — d3tt or
T = T exp (9n 2 /2nDC 3Tl ) (4 13)
When 371 = 0, the atomic magnets, responsible for the paramagnetic
properties of the material, are rotating at random with thermal mo
tion; impressing a magnetic field on the material tends to line up the
magnets and reduce their thermal motion and so to * 'squeeze out"
THE FIRST LAW OF THERMODYNAMICS 31
their heat energy, which must go into translational energy of the at
oms (increased temperature) since heat is not removed in an adiabatic
process. Reciprocally, if a paramagnetic material is magnetized,
brought down to as low a temperature as possible and then demagne
tized, the material's temperature will be still further reduced. By
this process of adiabatic demagnetization, paramagnetic materials
have been cooled from about 1°K to less than 0.01 °K, the closest
to absolute zero that has been attained.
The Second Law of
Thermodynamics
Once it had been demonstrated that heat is a form of energy, the
proof of the first law of thermodynamics became merged with the
proof of the conservation of energy. The experimental authentication
of the second law is less direct; in a sense it is evidenced by the suc
cess of thermodynamics as a whole. The enunciation of the second law
is also roundabout; its various paraphrases are many and, at first
sight, unconnected logically. We could introduce the subject by asking
whether there exists an integration factor for the imperfect differen
tial dQ, or by asking for a quantitative measure of the difference be
tween the quasistatic process leading from 1 to a in Fig. 41 and the
more rapid process of placing the gas immediately in contact with a
heat reservoir at temperature T a , or else by asking whether 100 per
cent of the heat withdrawn from a heat reservoir can be converted into
mechanical work, no matter how much is withdrawn. We shall start
with the last question and we find that in answering it we answer the
other two.
Heat Engines
Chemical or nuclear combustion can provide a rough equivalent of
a constant temperature heat reservoir; as heat is withdrawn more can
be provided by burning more fuel. Can we arrange it so this continu
ous output of heat energy is all continuously converted into mechanical
work? The second law of thermodynamics answers this question in
the negative, and provides a method of computing the maximum frac
tion of the heat output which can be changed into work in various cir
cumstances.
At first sight this appears to contradict a result obtained in Chap
ter 4. There it was pointed out that a system, with internal energy
that is a function of temperature only (such as a perfect gas or a per
fect paramagnetic material), when placed in contact with a constant
temperature heat source, can isothermally transform all the heat it
withdraws from the reservoir into useful work, either mechanical or
32
THE SECOND LAW OF THERMODYNAMICS
33
electromagnetic. The trouble with such a process is that it cannot
continue to do this indefinitely. Sooner or later the pressure gets too
low or the tension gets greater than the elastic limit or the magnetic
material becomes saturated, and the transformer's efficiency drops
to zero. What is needed is a heat engine, a thermodynamic system
that can operate cyclically, renewing its properties periodically, so it
can continue to transform heat into work indefinitely.
Such an engine cannot be built to run entirely at one temperature,
that of the heat source. If it did so the process would be entirely iso
thermal, and if we try to make an isothermal process cyclic by revers
ing its motion (compressing the gas again, for example) we find we are
taking back all the work that has been done and reconverting it into
heat; returning to the start leaves us with no net work done and all the
heat given back to the reservoir. Our cycle, to result in net work done
and thus net heat withdrawn, must have some part of it operating at a
lower temperature than that of the source. And thus we are led to the
class of cyclical operations called Carnot cycles.
Carnot Cycles
A Carnot cycle operates between two temperatures, a hotter, T^,
that of the heat source, and a colder, T c , that of the heat sink. Any
sort of material can be used, not just one having U a function of T
only. And any pair of mechanical variables can be involved, P and V
or J and L or 3C and arc (we shall use P and V just to make the dis
cussion specific). The cycle consists of four quasistatic operations:
an isothermal expansion from 1 to 2 (see Fig. 51) at temperature T^,
FIG. 51. Example of a Carnot cycle, plotted in the PV
plane.
34 THERMODYNAMICS
withdrawing heat aQ 12 from the source and doing work aW 12 (not nec
essarily equal to AQ 12 ); an adiabatic expansion from 2 to 3, doing fur
ther work aW^ but with no change in heat, and ending up at tempera
ture T c ; an isothermal compression at T c from 3 to 4 requiring work
AW 34 = AW 43 to be done on the system and contributing heat AQ 34
= AQ 43 to the heat sink at temperature T c , ending at state 4, so placed
that process 4 to 1 can be an adiabatic compression, requiring work
AW 41 = AW 14 (aQ 41 = 0) to be done on the system to bring it back to
state 1, ready for another cycle (Fig. 51). This is a specialized sort
of cycle but it is a natural one to study and one that in principle should
be fairly efficient. Since the assumed heat source is at constant tem
perature, part of the cycle had better be isothermal, and if we must
"dump" heat at a lower temperature, we might as well give it all to
the lowest temperature reservoir we can find. The changes in temper
ature should thus be done adiabatically.
This cycle, of course, does not convert all the heat withdrawn from
the reservoir at T^ into work; some of it is dumped as unused heat
into the sink at T c . The net work done by the engine per cycle is the
area inside the figure 1234 in Fig. 51, which is equal to AW 12 + AW 23
+ AW 34 + AW 41 = AW^ + AW 23  AW 43  AW 14 and which, according to
the first law, is equal to AQ 12 + AQ 34 = AQ 12  AQ 43 . The efficiency 77
with which the heat withdrawn from the source at Tjj is converted into
work is equal to the ratio between the work produced and the heat with
drawn.
_ AW 12 + AW 23  AW 43  AW 14 _ AQ^  AQ 43 _ AQ 43
71 AQ 12 AQ 12 AQ 12 V. 1 '
We note that, since all the operations are quasistatic, the cycle is re
versible; it can be run backward, withdrawing heat AQ 43 from the
reservoir at temperature T c and depositing heat AQ 12 in the reser
voir at Tfo, requiring work AQ 12  AQ 43 to make it go.
There are a large number of Carnot cycles, all operating between
Tfo and T c ; ones using P and V to generate work, involving different
substances with different equations of state; ones using ac and arc to
produce magnetic energy, using different paramagnetic substances;
and so on. One way of stating the second law is to say that all Carnot
cycles operating between the temperatures T n and T c have the same
efficiency. Another way is to say that no engine y or combination of en
gines, operating between a maximum temperature T^ and a minimum
temperature T c can be more efficient than any Carnot cycle operating
between these temperatures.
Statements of the Second Law
To show that these statements are equivalent we shall show that if
we could find a cycle of greater efficiency than a Carnot cycle, oper
THE SECOND LAW OF THERMODYNAMICS
35
ating between Th and T c , we can combine them to obtain a perfectly
efficient engine, thus more efficient than either. Figure 52 shows
how this can be done. We assume that the less efficient one is the
assumed
moreefficient
engine
AQ
AQ 12
standard
Carnot
engine
FIG. 52. Carnot engine (reversed) driven by an engine
assumed more efficient; the combination would
make a perfect engine, which is impossible.
"standard" one described in Eq, (51). We shall run this backward,
requiring net work AQ 12  AQ 43 to take heat AQ 43 from the lower tem
perature and depositing heat AQ 12 at the upper. We adjust the pre
sumed better engine so its exhaust AQ" at T c is equal to AQ 43 , the
same amount that the first engine withdraws. The amount of heat AQ'
it withdraws at T^ must be larger than AQ^ if its efficiency
1  (AQ"/AQ') is to be larger than the value 1  (AQ 43 /AQ 12 ) for the
"standard" engine. We now use this better engine to run the "stand
ard" one in its reversed cycle. Actually there will be work left over,
an amount (AQ'  AQ 43 )  (AQ 12  AQ 43 ) = AQ'  AQ 12 , which can be
used as we please. The combined engine thus withdraws a net heat
AQ'  AQ 12 from the upper heat reservoir, dumps no net heat into the
lower, and produces net work AQ'  AQ 12 ; it is a perfect engine. Thus
a contradiction of the first statement above leads to a contradiction of
the second statement. If there can be no engine more efficient than a
Carnot cycle, then all Carnot cycles (between T^ and T c ) must have
the same efficiency.
We can see now that still another way of stating the second law is
as follows: It is impossible to convert, continuously, heat from a res
ervoir at one temperature T^ into work, without at the same time
transferring additional heat from T^ to a colder temperature T c .
This way of stating it is called Kelvin's principle. Still another way
36 THERMODYNAMICS
is to state that it is impossible to transfer, in a continuous manner,
heat from a lower temperature reservoir to one at higher temperature
without at the same time doing work to effect the transfer. This is
called Clausius' principle. Its equivalence to the other three state
ments can be demonstrated by manipulating the combination of Fig.
52; by running it backward and adjusting the assumed better engine
so that AQ'  AQ" = AQ 12  AQ 43 and thus AQ"< AQ 43 and AQ'>AQ 12 ,
for example.
The Thermodynamic Temperature Scale
If all Carnot cycles operating between the same pair of tempera
tures, Th and T c , have the same efficiency, this efficiency must be
simply a function of T^ and T c :
V = l *(T h ,T c ); * (T h ,T c ) = (AQ 43 /AQ 12 ) (52)
The ratio of the heat dumped to that withdrawn must be the same for
all these cycles. To find how ^ depends on the temperatures T^ and
T c , as measured by a thermometer, we break up a Carnot cycle into
two cycles, each using the same material, as shown in Fig. 53. The
upper one takes heat AQ 12 from the upper reservoir at a temperature
6ft (on the scale of the thermometer we are using) does work AQ^ 
AQ 65 , and delivers heat AQ 65 to an intermediate reservoir at a meas
ured temperature 9 m . This reservoir immediately passes on this heat
AQ 65 to the second engine, which produces work aQ 65  AQ 43 and de
livers heat AQ 43 to a reservoir at temperature 9 C as measured on
our thermometer. The combination, which produces a total work of
AQ 12  AQ 43 , is thus completely equivalent to a single Carnot cycle,
using the same material and operating between #h and 9 C on our
scale, withdrawing AQ 12 from the upper, exhausting AQ 43 at the lower,
and doing work AQ 12  AQ 43 . Therefore, according to Eq. (52), the
efficiencies % and r\\ of the two component cycles and the efficiency
?7 C of the combination, considered as a single engine, are related as
follows :
T? u =l*(0 h >*m>; *^h,0m) = Hf
l^c = * ( ^ 6 c) = ^ = ^h,U*Mc) (53)
For the equation relating the three values of the function ^, for
the three pairs of values of measured temperature 9, to be valid, *
THE SECOND LAW OF THERMODYNAMICS
37
FIG. 53. Arrangement of two Carnot cycles so their com
bined effect is equivalent to one cycle between
the temperature extremes.
must have the functional form *(x,y) = [T(y)/T(x)] , where T(0) is
some single valued, monotonically increasing function of 0, the tem
perature reading on the thermometer used. For then ^(# m ,# c )*(£h^m)
will equal
38 THERMODYNAMICS
[T(0 c )/T(0 m )][T(0 m )/T(0 h )] =[T(9 C )/T(9 h )] = *(9 h ,0 c )
Therefore,
r'ia a\ AQ * 3 T( ^ c) AQ « AQ * 2 /c „x
* ( ^ c)= AQ^ = T5J5 ° r T^W (5 " 4)
We can thus experimentally determine the function T(d) by using
various Carnot cycles, all having the same upper temperature, read
as dfr on our thermometer, but each going to a different lower tem
perature, discarding a different amount of heat and thus doing differ
ent amounts of work. By measuring the common value of AQ 12 and
the different values of the heat discarded, AQ 43 , we can compute T(9 C )
as (aQ 43 /AQ 12 ) times the common value T(9\ i h If, for example, a cy
cle with lower reading 9^ has its discarded heat just 1/2 of AQ^,
then T(0 d ) = (l/2)T(0 h ).
The numerical values of 9& and all the # c 's were obtained from
the arbitrary scale of the particular thermometer used. It would seem
more appropriate to use T(#) itself, rather than 9, for a temperature
scale. We can use, for our upper reservoir, the temperature of melt
ing ice, and set T(# h ) = 273°. The value of T(# c ) for any colder tem
perature (such as the boiling point of oxygen, for example) is then
(AQ 43 /AQ 12 )x 273°, where AQ 12 and AQ 43 are the heats involved in the
Carnot cycle operating between the melting point of ice and the boiling
point of oxygen. Such a scale of temperature, as determined by meas
ured heat ratios for Carnot cycles, is called the thermodynamic scale,
and measurements given in this scale are in degrees Kelvin.
This now completes our series of definitions of temperature started
in Chapter 1. From now on temperature T will always be measured
in degrees Kelvin. In its terms, the efficiency of a Carnot cycle oper
ating between Th and T c (both measured in degrees Kelvin) is
AQ a, T r
1} = 1  *(T h ,T c ); *(T h ,T c ) = ^2.= ^ (55)
This is the maximum efficiency we can get from an engine that oper
ates between Th and T c .
Thus the second law is a sort of relativistic principle. The minimal
temperature at which we can exhaust heat is determined by the tem
perature of our surroundings, and this limits the efficiency of transfer
of heat into work. Heat at temperatures high compared to our sur
roundings is " high quality" heat; if we handle it properly a large por
tion of it can be changed into useful work. Heat at temperature twice
that of our surroundings (on the Kelvin scale) is already half degraded;
only half of it can be usefully employed. And heat at the temperature of
our surroundings is useless to us for getting work done. Even heat at
THE SECOND LAW OF THERMODYNAMICS
39
a million degrees Kelvin would be useless if the whole universe were
at this same temperature. Temperature differences enable us to pro
duce mechanical energy, not absolute magnitudes of average temper
ature.
FIG. 54. Reversible cycle (heavy line) simulated by a
combination of several Carnot cycles.
In principle we can build up a combination of Carnot cycles to sim
ulate any kind of reversible cycle, such as the one shown by the heavy
line in Fig. 54. In such a cycle, heat is taken on and given off at dif
ferent temperatures, none of the elementary processes being isother
mal or adiabatic. The maximum temperature reached is T^, for the
isothermal curve tangent to the top of the loop, and the minimum is
T c , for the lower tangent isothermal. The work produced is the area
within the heavy line. This cycle is crudely approximated by the five
Carnot cycles shown, with their isothermals and adiabatics as light
lines; a better approximation could be obtained with a large number of
Carnot cycles. The efficiency of subcycle 3 is greatest, because it
operates between the greatest spread of temperatures; the others have
less efficiency. Thus any cycle that takes in or gives off heat while
the temperature is changing is not as efficient as a Carnot cycle op
erating between the same maximum and minimum temperatures, i.e.,
which takes on all its heat at T^ and which gives up all its heat at T c .
Entropy
We notice that, for a Carnot cycle, the relationship between each
element dQ of heat taken on and the thermodynamic temperature T at
which it is taken on (or given off) is such that the integral of dQ/T
taken completely around the cycle is zero. The heat taken on at Th is
AQ 12 and the heat "taken on" at T c is the negative quantity AQ 34
= AQ 43 ; Eq. (55) states that the sum (AQ 12 /T h )  (aQ 43 /T c ) = 0.
A Thermal State Variable
Since any quasistatic, reversible cycle can be considered as a sum
of Carnot cycles, as in Fig. 54, we see that for any such cycle the in
tegral of the quantity £TQ/T around the whole cycle is zero. But for
any thermodynamic state function Z(x,y) (as in Fig. 61) the integral
of the perfect differential dZ around a closed path (such as ABA in
Fig. 61) is zero, as long as all parts of the path are reversible proc
esses; alternately any differential that integrates to zero around any
closed path is a perfect differential and its integral is a state function
of the variables x,y.
Therefore the quantity dS = dQ/T is a perfect differential, where
dQ is the heat given to the system in an elementary, reversible proc
ess and T is the thermodynamic temperature of the system during
the process. The integral of this perfect differential, S(x,y), is a state
variable and is called the entropy of the system. It is an extensive var
iable, proportional to n.
This result;, which is still another way of stating the second law,
can be rephrased to answer the first of the questions posed in the first
paragraph of Chapter 5. There is an integrating factor for dQ, if heat
dQ is absorbed in a reversible process; it is the reciprocal of the
thermodynamic temperature, defined in Eq. (55). The resulting per
fect differential dQ/T measures the change dS in the state variable
S, the entropy; and the difference &>  S x of entropy between equilib
rium states 1 and 2 i s computed by integrating dQ/T along any re
40
ENTROPY
41
FIG. 61. Paths in the xy plane for reversible processes
(solid lines) and for spontaneous processes
(dashed lines).
versible path between 1 and 2. On the other hand, there is no integrat
ing factor for tlQ for an irreversible process.
The entropy S is the extensive variable that pairs with T as V
does with P and sm with oc. The heat taken on by the system in a re
versible process is dQ = T dS, just as the work done by the system is
dW = P dV or ocdsni. Thus the equation that represents both the first
and second laws of thermodynamics is
dU = T dS  P dV + J dL + 5C dm + £ jbtj dn t +
l
(61)
for a reversible process. This equation, plus the empirical heat ex
pressions for heat capacity and the equations of state, constitutes the
mathematical model for thermodynamics, by means of which we can
predict the thermal behavior of substance in bulk.
Basic equation (61) can be integrated to obtain U, once we know
the dependence of T, P, J, etc., on S, V, L, etc., which we can take
to be the independent variables of the system. Of course we can, if we
choose, use P instead of V as one of the independent variables but, if
we desire to calculate U, Eq. (61) shows that the extensive variables
S, V, L, etc., are the "natural" ones to use. When expressed as func
tion of the extensive variables, U has the properties of a potential
function, for its partial with respect to one of the extensive variables
(V, for example) is equal to the corresponding thermodynamic "force,"
the related intensive function (P, for example). Thus
42 THERMODYNAMICS
T "(i) v „... ; P "~(«) T ,.. ; *"(») TV .J
Because Eq. (61) is a sum of intensive variables, each multiplied
by the differential of its corresponding extensive variable, we can ap
ply a trick devised by Euler to calculate U. Let the v extensive vari
ables appropriate for a system be X 1 ... ,Xj,; then the corresponding
intensive variables are Yj = dU/3Xj , all of which are uniform through
out the system at equilibrium. Now we increase the amount of material
by a factor A, keeping all the intensive variables Yj constant during the
change. The internal energy U for the new system is just A times the
U of the original system, and the extensive variables are also in
creased by A, so that
u(ax 1? ..., \x u ) = \u(x 1 ,.:.,x l/ )
Differentiating with respect to A on both sides and using the defini
tions of the intensive variables as partials of U, we have
£\J(\X 1 ,...,\X u )= £ (dU/SXjJXj = SY j X j = U(X 1 ,...,X i/ )
j=l j=l
Thus, in terms of our familiar variables,
U(S,V,L,3fTl,n 1 ,n 2 , ...) = ST  PV + JL + 3C9TI + £ ji^i + ••• (63)
i
This is Euler' s equation, which will be of use later. It may be con
sidered to be the basic equation of thermodynamics; all the rest may
be derived from it.
Reversible Processes
We are now in a position to be more specific about the adjective
reversible, which we first used for a cycle (such as a Carnot cycle)
and which we recently have been applying to processes. To see what
it means let us first consider a few irreversible processes. Suppose
a gas is confined at pressure P within a volume V of a thermally
insulated enclosure, as shown in Fig. 62. The gas is confined to V
by a diaphragm D; the rest of the volume, V x  V , is evacuated. We
then break the diaphragm and let the gas undergo free expansion until
it comes to a new equilibrium at volume V r This is a spontaneous
process, going automatically in one direction only. It is obviously ir
reversible; the gas would never return by itself to volume V .
Next suppose we place an object, originally at temperature T^, in
thermal contact with a heat reservoir at temperature T c , less than
ENTROPY
43
FIG. 62. The Joule experiment.
Th Here again the process is spontaneous; heat flows from the object
until it comes to equilibrium at temperature T c . This also is an ir
reversible process; it would take work (or heat from a reservoir at
Tjj) to warm the body up again.
We can thus define the adjective "reversible" in a negative way; a
reversible process is one that has no irreversible portion. To expand
the gas from V to V x reversibly we could replace the diaphragm by
a piston and move it slowly to the right. During the motion, as the vol
ume is increased by dV, the gas is never far from equilibrium, and a
reversal of motion of the piston (so the volume decreases again by dV)
would bring the gas back to its earlier state. In such a case we could
retrace every part of the process in detail. Every reversible process
is quasistatic; not all quasistatic processes are reversible.
In Eq. (61) we pointed out that the integral of cfQ/T around a re
versible cycle is zero. If the cycle is irreversible the integral differs
from zero; the second law requires it to be less than zero. For ex
ample, suppose the irreversible cycle took in all its heat, an amount
AQ' at T^ and exhausted an amount AQ" all at T c . The efficiency of
such a cycle would have to be less than the value 1  (T c /Th) = 1 
(aQ 43 /aQ 12 ) for a Carnot cycle between the same temperatures. This
means that AQ' would have to be smaller than AQ 12 or AQ" would have
to be larger than AQ 43 , or both, so that the integral of <TQ/T around
the irreversible cycle would turn out less than that for the Carnot cy
cle, i.e., less than zero. The argument can be generalized for all
closed cycles. Thus another way of stating the second law is that for
all closed cycles the integral around the cycle,
f (dQ/T) <
(64)
where the equality holds for reversible cycles and the inequality is
for irreversible ones. Since dS is measured by the value of dQ/T
when the process is reversible, we also have that
dS > dQ/T
(65)
44 THERMODYNAMICS
where again the equality holds for reversible processes, inequality
for irreversible ones.
Irreversible Processes
For example, suppose the dotted line from C at D in Fig. 61 rep
resents a spontaneous process, during which no heat is absorbed or
given up (as is the case with the free expansion of the gas in Fig. 62),
so that / (dQ/T) = for the dotted line. Since the process is irrevers
ible, / dS = Sd  Sq must be larger than / (ffQ/T) = 0. In other words,
during a spontaneous process taking place in a thermally isolated sys
tem, the entropy always increases.
The statement is not so simple when the spontaneous process in
volves transfer of heat, as with the irreversible cooling of a body from
Th to T c mentioned above [see also the discussion four paragraphs
below Eq. (41)]. In this case the body loses entropy and the reservoir
gains it. If the heat capacity of the body is C v , a constant, the heat
reservoir gains C v (Th  T c ) and, since the reservoir is always at T c ,
its gain in entropy is C v [(Th  T c )/T c ]. The loss in entropy of the
body is not much harder to compute. During its spontaneous discharge
of heat to the reservoir, and before it comes to uniform temperature
T c , the body is not in equilibrium, so dS does not equal dQ/T. How
ever we can devise a quasistatic process, placing a poor heat conduc
tor between the body and the reservoir, so the heat flows into the res
ervoir slowly and the body, at any time, will have a nearly uniform
temperature T, where T starts at T^ and gradually drops to T c at
the end. The loss of entropy from the body is thus the integral of
C v (dT/T), which is C v ln(Th/T c ) if C v is constant. This is always
smaller than C v [(Th  T c )/T c ] , the entropy gained by the reservoir,
although the two approach each other in value as T^ approaches T c .
Thus, although the entropy of the body decreases during the sponta
neous cooling of the body, the total entropy of body and reservoir
(which we might call the entropy of the universe) increases by the
amount
S = C v x C v ln(l+x) =C v (x 2  X 3 + ix 4  J
where x = [(T^  T c )/T c ] , which is positive for all values of x > 1.
Thus the statement at the end of the previous paragraph can be
generalized by saying that in a spontaneous process of any kind, even
if the entropy of the body decreases, the entropy of some other system
increases even more, so that the entropy of the universe always in
creases during an irreversible process. This, finally, is the answer
to the second question at the beginning of Chapter 5; the measure of
the difference between a reversible and an irreversible process lies
in the entropy change of the universe.
ENTROPY 45
Entropy is a measure of the unavailability of heat energy. The en
tropy of a certain amount of heat at low temperature is greater than it
is at high temperature, loosely speaking. Alternately, entropy meas
ures the degree of disorganization of the system. Irreversible proc
esses increase disorder, increase the amount of low temperature
heat, and thus increase the entropy of the universe. Reversible proc
esses, on the other hand, simply transfer entropy from one body to
another, keeping the entropy of the universe constant. A few examples
will familiarize us with these ideas.
Entropy of a Perfect Gas
The entropy of n moles of a perfect gas of point atoms, for which
U = (3/2)nRT and PV = nRT, may be determined by integration of Eq.
(61), which is for this case
T dS = dU + P dV or dS = (nR/T) dT + (nR/V) dV
SO
S  nR In [(T/T ) 3 / 2 (V/V )] + S
where S = S when T = T and V = V . Increase in either T or V in
creases the entropy of the gas. Instead of T and V (and n), S and V
(and n) can be used as independent variables, in which case
T = To $f e2(SS )/3nR; p . ^(^f .!» *>/>«*
These formulas immediately provide us with the dependence of T and
P on V for an adiabatic process. For a reversible adiabatic process,
dS = CTQ/T = 0, so S is constant, which is why partials for such proc
esses are labeled with subscript S.
We can also use Euler's equation (63) to calculate the chemical
potential per mole of point atoms. For this system, U = TS  PV + /in,
the atoms being all of one kind. Inserting the expressions for U, PV,
and TS in terms of T, V, and n and dividing by n, we find that
M  Ts + RT InLe 5 / 2 (V /V)(T /T) 3 / 2 ] (66)
where e is the base of the natural logarithms (In e = 1) and s
= (So/n,,) is the entropy per mole at T , V , and n . Therefore we can
use /x for an independent variable and obtain, for example,
V = V (T /T)3A exp (   \  £)
46 THERMODYNAMICS
The function U(S,V,n) is (nb /V 2/s ) e 2s/3nR , where constant b equals
T (V /n ) 2/3 e~ 2s o/ 3R , independent of S, V, and n.
The Joule Experiment
Let us return, for a page or two, to the irreversible, free expansion
process pictured in Fig. 62. Initially the gas is confined to volume V
and is at temperature T ; after the diaphragm has broken, the gas ex
pands spontaneously and eventually settles down to equilibrium in vol
ume V x at a new temperature T r We should like to be able to compute
T x and also to calculate the increase in entropy during the process for
any sort of gas, not just for a perfect gas. What we need to know to do
this is the nature of the final equilibrium state; we then can use the ap
propriate version of Eq. (61),
dU= TdS  P dV (67)
to integrate from initial to final state via a reversible path.
Equation (61), in its various forms, can represent any reversible
process, therefore can be used to compute the difference of value of
any state variable between any initial and final states. In the free
expansion case now under study, the actual process is far from re
versible, but it starts and stops with equilibrium states and we can
compute the difference between start and finish without bothering to
learn what the system did inbetween.
In the free expansion case, since the system is thermally insulated
during the process and no work is done, the internal energy must be
the same at the finish as it was at the start. Thus the change of value
of any state variable caused by free expansion may be computed by
integrating its rate of change with respect to V, at constant U. How
this partial may be found, by using Eqs. (36) and (37) to manipulate
Eq. (67), goes as follows.
First we express dS and dU in terms of their partials and the dif
ferentials of the independent variables T and V and equate coeffi
cients of these differentials:
5), «♦*{&)> ($,[(8)
SO
/as_\ = l/au\ = £v. (dS\ = l"f9U
laT/_ T\3T/ V T ' \8V/ T tLW
+ P
T
+ P
T
dV
(68)
Now apply the highly useful equation (37), on these partials of S,
ENTROPY 47
TL8V\3T/ V J T tL3T\8V/ t J v T\3T/ v T 2 l\dV 1
+ P
T
or
iGSHffL ♦**(»),
We also see that
OC v /8V) T = T
3T
(!I) T ] =T0 2 P/8T 2 ) V (610)
for any substance having only V and T as variables.
From these relationships we can compute the change of T with V
at constant U, by using Eq. (44) as well as Eq. (36) again:
This can be integrated if we know the empirical formulas for C v and
for P in terms of T and V. For example, for a perfect gas,
P = nRT/V, so P  T(3P/8T) V = and therefore (dT/dV)\j = 0. A
perfect gas comes to equilibrium, after a free expansion in an insu
lated container, with no change in temperature. Joule first proposed
and carried out measurements on such processes, to see how closely
actual gases behave like perfect gases. For monatomic gases the
state function n(3T/8V)u (which is called the Joule coefficient) is less
than 0.001°K. moles per m 3 .
For a gas satisfying the Van der Waals equation (32), having a heat
capacity C v that is independent of V,
 (£).
an"
so
an'
\dVJu V 2 C V (T)
which is small because a is small for most gases. This means that
we can consider C v constant over the limited range of temperature
involved, in which case the total temperature change caused by free
expansion is
48 THERMODYNAMICS
'■vsftH)
which gives the small change in T during the expansion at constant U
for a Van der Waals gas. Since V x > V the temperature drops during
the process, although the drop is small. During the expansion a small
amount of the molecular kinetic energy must be lost in doing work
against the small attractive forces between the molecules.
Entropy of a Gas
To compute the change of S with V at constant U for any sub
stance we need only note that the equation T dS = dU + P dV gives us
directly
0S/9V)xj=P/T (613)
which, for a perfect gas, results in S x  S = nR ln^/V,,) [which could
have been obtained from Eq. (65)] . For a Van der Waals gas,
(dS\ nR an 2
VaV/u Vnb V 2 T(V)
which can be integrated after we have substituted for T from Eq.
(611) (change T l9 V l into T, V and solve for T as a function of V,
V , T , then substitute this for the T in the term an 2 /V 2 T and inte
grate). However, as we found from Eq. (612), T changes but little
during the free expansion, so little error is made by setting T = T
in the small term (an 2 /V 2 T). Thus a good approximation to the change
in entropy of a Van der Waals gas during change from V to V x at
constant U is
**— *(I£S) $«*)
which should be compared with the entropy change for the perfect gas.
The entropy increases in both cases, as it must for a spontaneous
process of this sort. Since the system is insulated during the process,
the change represents an increase in entropy of the universe.
If we were to go from state to state 1 by a reversible process,
the increase in entropy of the gas would be the same as the value we
have just calculated. But, to offset this, some other system would lose
an equal amount, so the entropy change of the universe would be zero.
For example, a reversible change from V , T to V lf T for a perfect
gas would be an isothermal expansion, replacing the diaphragm by a
piston and moving the piston slowly from V to V,. The gas would do
work W = nRT ln^/Vo) and would gain entropy [see Eq. (69)]
ENTROPY
49
AS = fj 1 OS/8V) T dV = nR In (V^Vg)
For
T at
which is the same as that gained during the irreversible process
the isothermal process, however, we have a heat reservoir at
tached, which loses the amount of entropy that the gas gains, so the
change in entropy in the universe is zero in this case.
Entropy of Mixing
As a final example, illustrating the connection between entropy and
disorder, we shall demonstrate that mixing two gases increases their
combined entropy. As shown in Fig. 63, we start with an moles of a
gas of type 1 (such as helium) in a volume aV at temperature T and
FIG. 63. Arrangement to illustrate spontaneous mixing of
two different gases.
pressure P = cmRT /aV on one side of a diaphragm and (1  a)n
moles of gas of type 2 (such as nitrogen) in volume (1  ot)V on the
other side of the diaphragm, in equilibrium at the same temperature
and pressure. We now destroy the diaphragm and let the gases spon
taneously mix. According to our earlier statements the entropy should
increase, since the mixing is an irreversible process of an isolated
system. The total internal energy U of the combined system finally
has the same value as the sum of the U's of the two parts initially.
What has happened is that the type 1 gas has expanded its volume from
aV to V and type 2 gas has expanded from (la)V to V.
To compute the change of entropy, we use (8S/8V)u , which Eq.
(612) shows is P/T, which is nR/V for a perfect gas. Thus the
increase of entropy of gas 1 is cmR times the logarithm of the ratio
between final and initial volumes occupied by gas 1, with a similar ex
pression for gas 2. The total entropy increase, called the entropy of
mixing of the two gases, is
AS = nR
ot In
a
+ d
a ^{i^)
(615)
50 THERMODYNAMICS
which is positive for < a < 1. It is largest for a = 1/2, when equal
mole quantities of the two gases are mixed. We note that for perfect
gases, P and T of the final state are the same as those of the initial
state; for nonperfect gases P and T change somewhat during the
mixing (why?).
Entropy increase is to be expected when two different gases are
mixed. But what if the two gases are the same ? Does the removal of
a diaphragm separating two parts of a volume V, filled with one sort
of gas, change its entropy or not ? When the diaphragm is in place a
molecule on one side of it is more restricted in its travel than when
the diaphragm is removed; but this difference is unnoticeable macro
scopically. Does reinsertion of the diaphragm reduce the entropy
again? We must postpone the resolution of this paradox (called Gibbs'
paradox) until we treat statistical mechanics (Chapter 22).
Simple
Thermodynamic
Systems
We have already worked out the thermal properties of a perfect gas.
According to Eq. (69) its total energy is independent of volume for,
since P = nRT/V,
(3U/9V) T = T(3P/8T) V  P =
We note in passing that the T in Eq. (69) is the thermodynamic tem
perature, so that the T in the perfect gas law PV = nRT is likewise
the thermodynamic temperature. Thus our definition of a perfect gas
must include the statement that the T in the equation of state (31) is
the thermodynamic temperature.
The Joule Thomson Experiment
To see how nearly actual gases come to perfect gases in behavior
we can use the free expansion illustrated in Fig. 62. The measure
ment of temperature change in free expansion is called Joule's exper
iment and the partial derivative which is thus measured, n(9T/aV)jj,
is called the Joule coefficient of the gas. We note from Eqs. (6 10) and
(611) that the Joule coefficient for a perfect gas is zero and that for
a gas obeying Van der Waal's equation it is an 2 /V 2 C v , a small quan
tity for most gases.
As with the transformation of heat into work, we can devise a con
tinuous process corresponding to free expansion, as illustrated in Fig.
71. Gas is forced through a nozzle N by moving pistons A and B to
maintain a constant pressure difference across the nozzle (or we can
use pumps working at the proper rates). The gas on the highpressure
side is at pressure P and temperature T ; after going through the
nozzle it settles down to a pressure P x and temperature T r Suppose
we follow n moles of the gas as it goes through the nozzle, starting
from a state of equilibrium at P , T and ending at the state of
equilibrium 1 at P lt T 2 . We cannot follow the change in detail, for the
process is irreversible, but we can devise a reversible path between
51
52
THERMODYNAMICS
B
FIG. 71. The Joule Thompson experiment.
and 1 [or, rather, we can let Eq. (67) find a reversible path for us]
which will allow us to compute the difference between states and 1.
In particular, we can compute the temperature difference T x  T in
order to compare it with the measured difference.
This process differs from free expansion because the energy U
does not stay constant; work P V is done by piston A in pushing the
gas through the nozzle, and work P 1 V 1 is done on piston B by the time
the n moles have all gotten through. Thus the net difference in inter
nal energy of the n moles between state and state 1 is U x  U
= P V  PjVj. Instead of U remaining constant during the process,
the quantity
H = U + PV
(71)
is the same in states 1 and 0. This quantity, called the enthalpy of
the gas, is an extensive state variable.
The experiment of measuring the difference T x  T , when all parts
of the system shown in Fig. 71 are thermally insulated, is called the
JouleThomson experiment, and the relevant change in temperature
with pressure when H is kept constant, (8T/3P)h, is the Joule
Thomson coefficient of the gas. Experimentally, we find that for ac
tual gases, the temperature increases slightly at high temperatures;
at low temperatures the temperature T x is a little less than T . The
temperature at which (3T/9P) H = is called the Joule Thomson in
version point. Continuous processes, in which a gas, at a temperature
below its inversion point, is run through a nozzle to lower its temper
ature, are used commercially to attain low temperature. We use work
P V  PiV\ to cool the n moles of gas, so Clausius' principle is not
contradicted.
To compute the Joule Thomson coefficient we manipulate the equa
tion for H, or rather its differential,
SIMPLE THERMODYNAMIC SYSTEMS
53
dH = dU + P dV + V dP = T dS + V dP
(72)
as we did in Eq. (67) to obtain Eq. (610). First we note that the
change in heat T dS in a system at constant pressure equals dH, so
that Cp, the heat capacity of the system at constant pressure, is
(8H/8T)p, in contrast to Eq. (44). Just as internal energy U can be
called the heat content of a system at constant volume, so enthalpy H
can be called its heat content at constant pressure. In passing, we can
combine Eqs. (46) and (69) to express the difference between Cp and
C v for any system in terms of partials obtainable from its equation of
state ,
C p  C v = T(aV/8T) p (3P/dT) v
Next we manipulate Eq. (72) as we did Eq. (67), to obtain
p  — p  \aP/ T .vojt
(73)
/as_\ = i/aH\ =^p. (&L\ = J_r/an\ i = /av\
\BT) n tKbtL t' VaP/ T t LUp/t J V9T/
p
(74)
and
(3C p /8P) T = T0 2 V/3T 2 ) p
From this we can compute the Joule Thomson coefficient and the
change in entropy during the process,
(i) H = iS^ = ^[ v  T (f?) p ] and (S) H =
f
(75)
For a perfect gas V = T(3V/8T) p , so that (3H/aP) T = and the Joule
Thomson coefficient (3T/8P)jj is also zero; no change in temperature
is produced by pushing it through a nozzle. The change in entropy of a
perfect gas during the process is the integral of V/T = nR/P
with respect to P,
AS = nR hHPo/Pj
(76)
Since P > P x this represents an increase in entropy, as it must.
For a gas obeying Van der Waal's equation we can find (3V/8T) p
by differentiating the equation of state and manipulating,
2an 2
(V  nb) dP +
nRT
Vnb
()
\8T/ T
(V  nb)
X 
V 3
2an ^V
V 3
(V  nb)
nb) 2
dV = nR dT
RT
(V
nb) (l +
2an \
RTV/
54 THERMODYNAMICS
so that
(i) H J; (if  nb )
since a and b are small quantities. For this same reason T and
thus C p do not change much during the process, and we can write
Wo^(l%)(PoP>)
Since P > P x this predicts an increase in temperature during the
Joule Thomson process if T > 2a/Rb, a decrease if T < 2a/Rb,
the inversion temperature being approximately equal to 2a/Rb.
The corresponding increase in entropy during the process can be
computed, using the same approximations as before,
AS^n R ln(^)^(P P 1 )(l^) (77)
which is to be compared with the result of Eq. (76) for a perfect gas.
Black Body Radiation
Let us now turn to a quite different sort of system, that called
blackbody radiation, electromagnetic radiation in equilibrium with
the walls of an enclosure kept at temperature T. This is the radia
tion one would find inside a furnace with constanttemperature walls.
It consists of radiation of all frequencies and going in all directions,
some of it continually being absorbed by the furnace walls but an equal
amount continually being generated by the vibrations of the atoms in
the walls. This is a special kind of system, with special properties.
In the first place the energy density e of the radiation, the mean
value of (1/2) & • £) + (1/2) 3C • (B, depends on the temperature but is
independent of the volume of the enclosure. If the volume inside the
furnace is enlarged, more radiation is generated by the walls, so that
the energy density e remains the same. Therefore the total electro
magnetic energy within the enclosure, Ve(T), is proportional to the
volume. This is in contrast to a perfect gas, for which U at a given
temperature is constant, independent of volume. If the enclosure is
increased in volume the density of atoms diminishes and so does the
energy density; for radiation, if the volume is increased more radia
tion is produced to keep the density constant. We can, of course, con
sider the radiation to be a gas of photons, each with its own energy,
but the contrast with atoms remains. Extra photons can be created to
fill up any added space; atoms are harder to create.
SIMPLE THERMODYNAMIC SYSTEMS 55
In the second place the radiation pressure, the force exerted per
unit area on the container walls, is proportional to the energy density.
In this respect it is similar to a perfect gas of point atoms, but the
proportionality constant is different. For the gas [see Eq. (24)]
P = (2/3)[N<K.E.> tran /V] = (2/3 )e, where e is the energy contained
per unit volume; for radiation it is (1/3 )e. The difference comes from
the fact that the kinetic energy of an atom, (l/2)mv 2 , is onehalf its
momentum times its velocity. For a photon, if its energy is ha;, its
momentum is (tiaj/c) where c is its velocity, (a;/27j) its frequency,
and h = 27Tn is Planck's constant; therefore the energy of a photon is
the product of its velocity and its momentum, not half this product.
Since pressure is proportional to momentum, e = 3P for the photon,
= (3/2)P for the atom gas.
To compute U, S, and P as functions of T and V we start with the
basic equation again, dU = T dS  P dV, inserting the appropriate ex
pressions, U = Ve(T) and P  (l/3)e(T),
T ds " T (ll) v aT + T (lv) T av ■ "" * p av
1 + e dV + e dV
or
(d§_\ = V /de\ (dS_\ = 4 e_
V3T/ V T \dT/' \dV) T 3 T
Applying Eq. (310) we obtain a differential equation for e(T),
1 (d±\ = i 1 (de_\ _ i _£_ de_ = 4 e_
TVdT/ 3 T\dT/ 3 T 2 ° r dT T
which has a solution e(T) = aT 4 , so
U = aVT 4 ; P=aT 4 ; S = aVT 3 (78)
o o
The equation for the energy density of blackbody radiation is called
Stefan's law and the constant a is Stefan's constant. In statistical
mechanics [see Eq. (259)] we shall evaluate it in terms of atomic
constants.
We see that the energy of blackbody radiation goes up very rapidly
with increase in temperature. Room temperature (70° F) is about
300° K. At the temperature of boiling water (373° K) the energy density
of radiation is already 2 1/2 times greater; at dull red heat (920° K) it
56 THERMODYNAMICS
is 100 times that at room temperature. At the temperatures encoun
tered on earth the pressure of radiation is minute compared to usual
gas pressures. At temperatures of the center of the sun (10 7o K) the
radiation pressure supports more than half the mass above it.
Reference to Eq. (63), U = ST  PV + jiin shows that the chemical
potential of blackbody radiation is
jll = (l/n)(U  ST + PV) = (l/n)(aVT 4   aVT 4 +  aVT 4 J =
which is related to the freedom with which the number of moles of
photons adjusts its value to keep the energy density constant [see Eq.
(252)] . In this matter, also, photons are a special kind of gas.
Paramagnetic Gas
Finally let us work out the behavior of a paramagnetic, perfect gas,
Here the two equations of state are P = nRT/V and 5C = Tsm/nD.
The heat capacity at constant V and 371 is a constant. C y3Tl = (3/2)nR
for a monatomic gas. The basic equation is
dU = T dS  P dV + JC d3TC
Three independent variables must be used; we first use T, V, and 971.
By methods that should be familiar by now we find
T (as\ = 3 nR /am
/as\ /_au\ _ 3C= _ T (dx\
\a37i/ TV \3gn/ TV VaT/o^
Other manipulations result in
(au/av) Tgll = (au/a97i) TV = o
Cpx= C V3 k + P() p  *(f?) x = fnR ♦ (^/nD)
Integrating the partials for U and S, we obtain
"V_/_T_\3/
VoAtJ
U = nRT; S = nR In
& +S « (7 " 10)
SIMPLE THERMODYNAMIC SYSTEMS 57
The "natural variables" in terms of which to express U are S, V, M
rather than T, V, 9H. To do this we express T in terms of S, V, 9H
and obtain
[see Eq. (65) et seq.] .
The extensive variables S, V, SfTC are less easy to measure than
are the intensive variables
P = 0U/3V) Sgrc ; T = OU/8S) V3TZ ; H = OU/83tt) sv
which might be called the "experimental variables." Expressing the
basic equation in terms of these (using P dV = (nRT/P)dP + nR dT,
for example) we have
TdS = (fnR + nD£)dTS5IdP252SdX ( 7 _ 12 )
For isothermal operation, dT is zero. The heat contributed to the
gas by the reservoir at temperature T, when the gas pressure is
changed from P to P x and the magnetic intensity is changed from
3C to flfCj, is
AQ 01 = nRT lnCPo/Pj + (nD/2T)(3C^  3Cj) (7 13)
Increase in pressure squeezes out heat (aQ < 0) as does an increase
in magnetization. At low temperatures a change in magnetic field pro
duces more heat than does a change of pressure.
The behavior of the system during an adiabatic process can be
computed by setting T dS = 0. The integrating factor is 1/T and the
integral of
(!?+■>¥)"
T 2
is
nD3C , nR ,„ n
dx = dP =
©(r[i(^f]='
If the magnetic field is kept constant (3C 1 = 3C ), the gas undergoes
ordinary adiabatic compression and T x is proportional to the two
fifths power of the pressure P 2 . Or, if the pressure is kept constant,
the temperature is related to the magnetic field by the formula
58 THERMODYNAMICS
3C» = T[(3C/T§) + (5R/D) In (VT )] (715)
At lowenough initial temperatures or highenough initial fields, so
that (3C /T ) 2 ^> 5R/D , the final temperature T l is approximately
proportional to the final magnetic intensity 3C X ; an adiabatic reduction
of 3C proportionally lowers the temperature.
Finally, we can have a process which is both adiabatic and isother
mal if we adjust the pressure continually so that, as the magnetic field
is changed adiabatically, the temperature is kept at a constant value
T (the volume then changes inversely proportional to P). The rela
tion between JC 1 and P x that will keep T constant is
P 1 = P exp
2RT 2 13L 3L t ;
(716)
As 3C X decreases, T? 1 will have to be increased exponentially to keep
T constant. In this process, mechanical work is used to demagnetize
the material.
Actual magnetic materials are not perfect gases, nor does their
magnetic equation of state have exactly the form of the simple Curie
law. In these cases one tries to find equations of state that do fit the
data and then tries to integrate Eq. (712). Failing this, numerical
integration must be used to predict the thermal properties of the ma
terial.
The
Thermodynamic
Potentials
In connection with Eq. (61) we pointed out that U, the internal en
ergy, when expressed in terms of the extensive variables S, V, L,
9TC, n^, etc., behaves like a potential energy, in that the thermodynamic
"forces," the intensive variables, are expressible as gradients of U,
(§)„ „■ — (f?L= (!?)„..= •
(81)
as was indicated in Eq. (62).
The Internal Energy
Moreover if we include irreversible as well as reversible proc
esses, Eqs. (41), (61), and (65) show that
dU = 3Q  P dV + /i dn + J dL + JC d3U + •••
< T dS  P dV + jll dn + J dL + JC dgn + ••• (82)
where the equality holds for reversible processes, the inequality for
irreversible ones. Consequently if all the extensive variables are
held constant (dS  dV = dL = ••• = 0) while the system is allowed to
come spontaneously to equilibrium, then every change in U will have
to be a decrease in value; it will only stop changing spontaneously
when no further decrease is possible; and thus at equilibrium U is
minimal. In other words, when the extensive variables are fixed, the
equilibrium state is the state of minimal U.
It is usually possible (although not always easy) to hold the mechan
ical extensive variables, V, L, n, arc, etc., constant, but it is more
difficult to hold entropy constant during a spontaneous process. How
ever if we allow the process to take place adiabatically, with ftQ =
59
60 THERMODYNAMICS
(and hold dV = dL = dn = ••• = 0), then U does not change during the
process and T dS ^ 0, so that S will reach a maximum value at equi
librium. If we had tried to keep S constant during the spontaneous
process we would have had to withdraw heat during the process,
enough to cancel out the gain in S during the process; this would have
decreased U, until at equilibrium U would be minimal, less than the
original U by the amount of heat that had to be withdrawn to keep S
constant.
When all the mechanical extensive variables are held constant, an
addition of heat to the system produces a corresponding increase in
U. We thus can call U the heat content of the system at constant V,
L, n, ••• , and can write the heat capacity
C vn ... = 0U/3T) vn ... (83)
the subscripts indicating that all the extensive mechanical variables,
which apply to the system in question, are constant. Moreover we can
apply Eq. (310) to the various intensive variables of Eq. (81), obtain
ing a very useful set of relationships,
til) =(m . (21) (*n)
(dT_\ = (dx\ _ /8P\ = (djA
W sv ... \3s; vgri ...' Van/gv... \sv; Sn ...
which are called Maxwell's relations. The relations, of course, hold
for reversible processes. They are used to compute the differences
in value of the various thermodynamic variables between two equilib
rium states.
Enthalpy
Although the extensive variables are the "natural" ones in which
to express U, they are often not the most useful ones to use as inde
pendent variables. We do not usually measure entropy directly, we
measure T; and often it is easier to keep pressure constant during a
process than it is to keep V constant. We should look for functions
that have some or all of the intensive variables as the "natural" ones.
Formally, this can be done as follows. Suppose we wish to change
from V to P for the independent variable. We add the product PV to
U, to generate the function H = U + PV, called the enthalpy [see Eq.
(71)] . The differential is
THE THERMODYNAMIC POTENTIALS
61
dH = P dV + V dP + dU
T dS + V dP + /! dn + 3C dm +
(85)
where again the equality holds for reversible processes, the inequality
for irreversible ones. If a system is held at constant S, P, n, ..., the
enthalpy will be minimal at equilibrium (note that P, instead of V, is
held constant). If H is expressed as a function of its "natural" coor
dinates S, P, n, ..., then the partials of H are the quantities
Us) Pn ... T; Up) Sn ... V; UJ SP ... M
and the corresponding Maxwell's relations are
(§L
' 3S /pn. :
/av
\3 arc/si;
L..(
ajc\
8P/,
S3TT
etc.
(86)
etc.
(87)
Geometrically, the transformation from U to H is an example of a
Legendre transformation involving the pair of variables P, V. Func
tion U(V), for a specific value of V (i.e., at point Q in Fig. 81) has
\>>
,x ] Q
f /
V
FIG. 81. Legendre transformation from U as a function
of V to H as a function of P
a slope dU/dV = P(V), which defines a tangent, HQ of Fig. 81,
which has an intercept on the U axis of H = U + PV. (We are not con
62 THERMODYNAMICS
sidering any other variables except P, V, so we can use ordinary de
rivatives for the time being.) Solving for V as a function of P from
the equation dU/dV = P, we can then express H as a function of
the slope P of the tangent line. Since dU = P dV and dH = dU +
P dV + V dP = V dP we see that dH(P)/dP = V(P). Thus enthalpy H
is the potential that has P as a basic variable instead of V.
Since any addition of heat T dS to a system when P, n, ... are held
constant causes a like increase of H, the enthalpy can be called the
heat content of the system at constant pressure.
The Helmholtz and Gibbs Functions
At least as important as the change from V to P is the change
from S to T. Adiabatic (constant S) processes are encountered, of
course, but many thermodynamic measurements are carried out at
constant temperature. The Legendre transformation appropriate for
this produces the Helmholtz function F = U  TS, for which
dF < S dT  P dV + J dL + \i dn + 3Cdgn+ ••• (88)
(I) vn ...* MLA 8S W ..." ""
The related Maxwell relations include
(i) Tn ... = (ff) Vn ... ; © TV ... = (!f) v:ni ... ; etc  (8 " 9)
Since the heat capacity C Vg]l ... is equal to T(aS/aT)yc JTl ... [see Eq.
(68)] , by differentiating these relations again with respect to T we
obtain a set of equations
(^ C V3nL ..  T (?)
T3TC" V01 'V
~ C ^)= T (^L etc  (8  10)
[d^l^^'ljy...
971
which indicate that the heat capacity is not completely independent of
the equations of state. The dependence of Cyan... on T is something
we must obtain directly by measurement; its dependence on the other
variables, V, arc, etc., can be obtained from the equations of state.
By the same arguments as before, the inequality of Eq. (88) shows
that for a system in which T, V, n, an, etc., are held constant, the
Helmholtz function F is minimal at equilibrium. If T is held con
stant, any change in F can be transformed completely into work, such
THE THERMODYNAMIC POTENTIALS 63
as P dV or 3C dsm, etc. Thus F is sometimes called the free en
ergy of the system at constant temperature. The Maxwell relations
(89) are particularly useful. For example, the second equation shows
that, since an increase of magnetic polarization increases the order
liness in orientation of the atomic magnets and hence entropy (a meas
ure of disorder) will decrease as 9TC increases, therefore the mag
netic intensity 3C required to produce a given magnetization arc will
increase as T is increased. Similarly we can predict that since rub
ber tends to change from amorphous to crystal structure (i.e., be
comes more ordered) as it is stretched, therefore the tension in a
rubber band held at constant length will increase as T is increased.
Another potential of considerable importance is the Gib bs function
G = U+PVTS=F + PV, having T and P for its natural coordi
nates, instead of S and V, and for which
dG < S dT + V dP + /i dn + J dL + JC d 201 + ••• (811)
The Gibbs function is a minimum at equilibrium for a system held at
constant T, P, n, 9TC, ... , a property that we will utilize extensively in
the next two chapters. The Maxwell relations include
(§L..=dL.. (f )„...= (§Lj * '»»
from which we can obtain the dependence of the heat capacity Cp^ ...
on the mechanical variables
fe c p^) T3rr ...=  T (S)p^ etc  (8  i3)
This process can be continued for all the mechanical variables,
obtaining a new potential each time, and a new set of Maxwell rela
tions. For example there is a magnetic Gibbs function G m = U 
TS + PV  3C am . From it we can obtain the following:
= (^k\ (<FL\ = _/M\ . etc
TP... \3T/ pac ...' Vaoc/ TP ... V3P/TJC...'
(814)
Finally there is what is called the grand potential £2, obtained by a
Legendre transformation from n to jj. as the independent variable,
which is useful in the study of systems with variability of number of
64
THERMODYNAMICS
particles, such as some quantum systems exhibit:
fi = U  TS  jLin; dO < S dT  P dV  n d^ + J dL +
from which the following Maxwell relations come:
/as \ = (dn_
Um/tv \ dT '
Vli
W/ TV ... \dVj
Tm
(815)
etc. (816)
Procedures for Calculation
A technique for remembering all these relationships can be worked
out for simple systems, such as those definable in terms of T and V,
with n constant. Here, when we use Euler's equation (63),
U = ST  PV + /in; H = U + PV = ST + /in
F = U  ST = PV + j^n; G  F + PV = /in (8 17)
The mnemonic device is shown in Fig. 82. It indicates the natural
variables for each of the four potentials (S and V for U, etc.) and the
arrows indicate the basic partial derivative relations, with their signs,
FIG. 82. Diagram relating the thermodynamic potentials
with their natural variables, their partials, and
the partials of these variables (Maxwell rela
tions) for a simple system.
such as (aH/3P) s = +V and (3F/3T)y = S. It also indicates the na
ture of the various Maxwell relations, such as (aS/3V) T = (3P/8T)y
THE THERMODYNAMIC POTENTIALS 65
or (3S/8P) T = (3V/3T)p; the arrows this time connect the numera
tor of the partial derivative with the subscript on it and the directions
of the arrows again indicate the sign.
Using this device we are now in a position to formulate a strategy
for expressing any possible rate of change of a thermodynamic vari
able in terms of the immediately measurable quantities such as heat
capacity and an equation of state, or else the empirically determined
partials (8V/dT)p and (3V/8P)x from which we could obtain an equa
tion of state. Either C v or C p can be considered basic (C p is the
one usually measured),
^ =©,(§),: Cp; (§)„(§)„ >•>
The relation between them is given in terms of partials from the
equation of state [see Eq. (73)] ,
C p = C v + T(8P/8T) V (8V/3T) p (8 19)
The various tactics which can be used to express an unfamiliar
partial in terms of an immediately measurable one are:
(a) Replacing the partials of the potentials with respect to their ad
joining variables in Fig. 82 by the related variables, such as
(3F/3T) V = S or (3U/3S) V = T, etc.
(b) Replacing a partial of a potential with respect to a nonadjacent
variable, obtainable from its basic equation, such as dF = S dT 
P dV, from which we can get (aF/3V)s = S(3T/aV) s  P, and
(3F/8S) p = S(3T/aS) p  P(3V/aS) p , etc.
(c) Using one or more of the Maxwell relations, obtainable from
Fig. 82.
(d) Using the basic properties of partial derivatives, as displayed
in Eqs. (39) and (310).
In terms of these tactics, the appropriate strategies are:
1. If a potential is an independent variable in the given partial,
make it the dependent variable by using (d) (this process is called
bringing the potential into the numerator) and then use (a) or (b) to
eliminate the potential. Examples:
1
(3U/3T) V C v
_/8T\ (3U/aV)T _ T (dS\ _ _P
\dV) v (8U/3T) V C v \dVj T C v
2. Next, if the entropy is an independent variable in the given par
66 THERMODYNAMICS
tial or in the result of step 1, bring S into the numerator and elimi
nate it by using (c) or Eq. (818). Examples:
(dP\ (dP/dT) v _t (dP\
\dS) v OS/8T) v C v \dT) Y
_ (BT\ QS/3V) T = _T /3P\
\dVJ s (3S/8T) V C v Ut/ v
3. If the measured equation of state partials have V in the numer
ator, bring V into the numerator in the result of steps 1 and 2 by us
ing (d). If the equation of state is in the form P = f(V,T), bring P into
the numerator.
The result of applying these successive steps will be an expression
for the partial of interest in terms of measured (or measurable) quan
tities. It is obvious that if other variables, such as 9TC and X instead
of V and P , are involved, a square similar to Fig. 82 can be con
structed, which would represent the relationships of interest, and pro
cedures 1 to 4 can be carried through as before.
Examples and Useful Formulas
As one example, we shall anticipate a problem discussed in Chap
ter 9. There we shall find it enlightening to plot the Gibbs function G
as a function of T for constant P. We can obtain such plots by inte
grating (3G/3T)p = S or, since S is not a directly measured quan
tity, by the double integration of (aS/8T) p = C p /T with respect
to T. In the case of a gas, where C v is a function of T alone, we
must then express Cp as a function of T and P, so it may be inte
grated easily. Thus several applications of the procedures outlined
above result in
(FG\ Cp C V (T) _ (dV\ (BP\ _C V [(8V/3T) p ] 2
\8T 2 /p T T \aTy p VaT/ v T (dV/3P) T
so
7 T fC v [OV/3T) D ] 2
G = G (P)  (T T )S (P)  / dT J \^  (g v /a p^
T T c
dT
(820)
Since S (P), the entropy at temperature T at the constant P is a
positive quantity; the plot of G against T for constant P has a nega
tive slope. Furthermore, since C v is positive and (3V/3P)t is neg
ative for all T and P, the slope of G becomes more negative as T
increases; the plot curves downward (see Figs. 92 and 96).
THE THERMODYNAMIC POTENTIALS 67
Finally, and in part to collect some formulas for useful reference,
let us work out again the entropy and then the thermodynamic poten
tials for a perfect gas of point atoms, from its equation of state PV
= nRT and its heat capacity C v = (3/2)nR. To find the entropy we use
(3S/3T) V = C v /T = 3nR/2T, and also (aS/9V) T = OP/8T) v = nR/V,
so that
S = fnR In (T/T ) + nR In (V/V ) + ns
as per Eq. (65). To find the Helmholtz function, and thence the other
potentials, we use (3F/8T) V = S and (3F/3V) T = P, so that
F = nRT  nRT In (T/T )  ns T  nRT In (V/V )
U = F + TS = nRT = nRT (V /V) 2/3 exp [2(S  ns )/3nR]
H = U + PV  nRT  nRT (T P/P T) 2 / 3 exp[2(S  ns )/3nR]
(821)
G = F + PV  (nR  ns ) T  nRT In (T/T ) + nRT In (V/V )
= njLL= fnRns W  nRT ln(T/T ) + nRT ln(P/P )
£1 = F  n/x = (nR  ns ^T  nRT In (T/T )  nRT In (V/V )  n/x
where we have utilized the fact that U = when T = to fix the value
of the constant of integration F . These can be checked with Eqs. (81)
to (815) for selfconsistency.
We note that the chemical potential per mole of the gas is a func
tion of the intensive variables T and P only and is thus independent
of n (as it must be). In fact we can see that all the thermodynamic
potentials are n times a function which is independent of n, for
V/V and T/T are both independent of n. This general property
of thermodynamic potentials must be true for any system involving a
single component (why?).
Changes of Phase
Every substance (except helium) is a solid at sufficiently low tem
peratures. When it is in equilibrium it has a crystalline structure
(glasses are supercooled liquids, not in equilibrium). As has already
been mentioned and will be discussed later in some detail, entropy is
a measure of randomness, so it will not be surprising to find that the
entropy of a perfect crystal vanishes at absolute zero. (This state
ment is sometimes called the third law of thermodynamics.)
The Solid State
To see how the entropy and the thermodynamic potentials of a solid
change as the temperature is increased, we shall utilize the simplified
equation of state of Eq. (36),
V = V (l+j3T kP); P = (j3A)T  [(VV )/kV ] (91)
Its heat capacities go to zero at zero temperature (see Fig. 31) and
C v rises to 3nR at high temperatures. The capacity Cp is related to
C v according to Eq. (73). A simple formula that meets these require
ments is
C v = [3nRT 2 /(S 2 + T 2 )]; C p = C v + (/3 2 V /k)T (92)
The formulas do not fit too well near T = 0, but they have the right
general shape and can be integrated easily. Constant 6 usually has a
value less than 100° K so that by room temperature C v is practically
equal to 3nR.
Using Eqs. (74) we compute the entropy as a function of T and P,
remembering that S = when T = and P = 0,
S = nR ln[l + (T 2 /0 2 )] + (B 2 V /k)T  /3V P (93)
68
CHANGES OF PHASE
69
This formula makes it appear that S can become negative at very low
temperatures if P is made large enough. Actually /3 becomes zero
at T = 0, so that at very low temperatures S is independent of pres
sure. At moderate temperatures and pressures Eq. (93) is valid;
curves of S as function of T for different values of P are shown in
Fig. 91.
FIG. 91. Solid lines plot entropy of a solid as a function
of T and P.
Next we use Eqs. (811) to compute the Gibbs function G as func
tion of T and P,
G = f nRT In fl + £j + 3nRT  3nR<9 tan" 1 (T/0)
 (V /3 2 /2k)T 2 + V p[l + j3T  kPJ + U (94)
where U is a constant of integration. Typical curves for this func
tion are plotted in Fig. 92 for different values of P. We note that G
is nearly constant at low temperatures, dropping somewhat at higher
temperatures, as indicated by Eq. (820).
Melting
If we add more and more heat (quasistatically) to the crystalline
solid, holding the pressure constant at some moderate value, its tern
70
THERMODYNAMICS
solid P = k/50
s
^V solid P = ,/mnn
solid p = io~ 5 /c " "^!*
I
I
I
I !
9
FIG. 92. Solid lines plot Gibbs function of a solid as a
function of T and P.
perature rises until finally it melts, turning into a liquid with none of
the regularities of the crystal. During the melting, addition of more
heat simply melts more crystal; the temperature does not rise again
until all is melted. The temperature T m at which the melting occurs
depends on the pressure, and the amount of heat required to melt 1
mole of the crystal L m is called the latent heat of melting (it also is
a function of the pressure).
We wish to ask how thermodynamics explains these facts, next to
find whether it can predict anything about the dependence of T m , the
temperature of melting, on P, and of the latent heat of melting L m
on any of the thermodynamic quantities.
The answer to the first question lies in the discussion of Eq. (811),
defining the Gibbs function. At any instant during the quasistatic proc
ess of heat addition, the temperature and pressure are constant; thus
the material takes up the configuration which has the lowest value of
G for that T and P. Below T m the G for the solid is less than the
G for the liquid; above T m the liquid phase has the lower value of G;
if the process is carried out reversibly, all the material must melt at
the temperature T m (P), at which the two G's are equal (for the pres
sure P).
If we could supercool the liquid to T = we would find its entropy
to be larger than zero, because a liquid is irregular in structure. Fur
thermore the entropy of the liquid increases more rapidly with tem
perature than does the S for the solid. Since (8G/8T)p = S, even
though at T = the G for the liquid is greater than the G for the
CHANGES OF PHASE 71
solid, it will drop more rapidly as T increases (see the dotted line of
Fig. 92) until, at T = T m the two G's are equal; above T m the liquid
has the lower G and is thus the stable phase. Thus thermodynamics
explains the sudden and complete change of phase at T m .
The second question, raised earlier, is partly answered by pointing
out that heat must be added to melt the material and that, at constant
T an addition of heat ttQ corresponds to an increase in entropy of the
substance by an amount tTQ/T . Thus the n moles of liquid, at the
melting point T m , has an entropy nL m /T m greater than the solid
at the melting point, where nL m is the heat required to melt n moles
of the solid (at the specified P). Thus a measurement of the latent
heat of melting L m enables one to compute the entropy of the liquid
at T m , in terms of the entropy of the solid (which is computed by in
tegration from T = 0), and further integration enables one to compute
S for the liquid, for T greater than T m , knowing the heat capacity
and the equation of state of the liquid.
ClausiusClapeyron Equation
To answer the rest of the second question we utilize the fact that
at the melting point the Gibbs function G s (T m ,P) for the solid equals
the Gibbs function Gi(T m ,P) for the liquid, no matter what the pres
sure P. In other words, as we change the pressure, the temperature
of melting T m (P), the Gibbs function G s for the solid and G\ for the
liquid all change, but the change in G s must equal the change in G^,
in order that the two G's remain equal at the new pressure. Refer
ring to Eq. (811) we see that this means that
dG s = " s s dT + v s dP = dG l = ~ s l dT + v l dP
or
(Vj  V s ) dP = (Si  S s ) dT m = (nL m /T m ) dT m
since the difference in entropy between a mole of liquid and a mole of
solid is equal to the latent heat divided by the temperature of melting.
Thus the equation relating T m to P is
dT m /dP = (T m /nL m )(Vi  V s ) (95)
which is the ClausiusClapeyron equation.
If the volume of the liquid is greater than the volume of the solid,
then an increase of pressure will raise the temperature of melting so
that, for example, if such a liquid is just above its melting point, an
increase in pressure can cause it to solidify. Vice versa, if the solid
is less dense than the liquid (as is the case with ice) an increase of
72 THERMODYNAMICS
pressure lowers the melting point and pressure can cause such a solid,
just below its melting point, to melt. Thus the fact that ice floats is
related to the fact that ice skating is possible; ice skates ride on a
film of water which has been liquefied by the pressure of the skate. In
general, since Vj differs but little from V s , the pressure must be
changed by several thousand atmospheres to change T m by as much
as 10 per cent.
Evaporation
If now heat is added to the liquid, its temperature will increase un
til another phase change occurs— the liquid evaporates. Here again
the temperature remains constant at the temperature of vaporization
T v until all the liquid is converted into vapor. To be sure we under
stand what has taken place let us examine the process in more detail.
We have tacitly assumed that the substance, first solid and then liquid,
is confined in a container which adjusts its volume V so that it exerts
a pressure P on all parts of the outer surface of the material. In
other words we have assumed that the volume V is completely filled
by the substance.
This may be difficult to do for the solid, but it is not hard to ar
range it for the liquid. We provide the container with a piston, which
exerts a constant force on the liquid and which can move to allow the
liquid to expand at constant pressure P, and we make sure the liquid
completely fills the container. In this case the liquid will stay liquid
while we add heat, until its temperature reaches T V (P), when it must
all be converted into gas (at a much greater volume but at the same
pressure) before additional heat will raise the temperature beyond
T v . The temperature of vaporization T V (P) is related to the pressure
by another ClausiusClapeyron equation,
dT v /dP = (T v /nLy)(V g  V X ) (96)
where here Ly is the latent heat of evaporation per mole of the mate
rial (at pressure P), Vj is the volume of the material as a liquid be
fore evaporation, and V g is its volume as a gas, after evaporation, at
T v and P. Since Vg is very much larger than V\, T v changes much
more rapidly with P than does T m .
However our usual experience is not with the behavior of liquids
that entirely fill a container, but with evaporation from the free sur
face of a liquid. When a liquid (or solid) does not completely fill a
container, some of the substance evaporates into the free space until
there is enough vapor there so that equilibrium between evaporation
and condensation is reached. This equilibrium is only reached when
the temperature of the liquid and vapor is related to the pressure of
CHANGES OF PHASE 73
the vapor in the free space above the liquid by the functional relation
ship we have been writing, T V (P), determined by Eq. (96). In the
case we are now discussing it is better to reverse the functional re
lationship and write that the vapor pressure P v is a function of the
temperature T, and that the equation specifying this relationship is
the reciprocal of (96),
dP v /dT = [nl^/TCVg  Vi)] (97)
This is the morefamiliar form of the ClausiusClapeyron equation.
The presence of another kind of gas in the space above the free
surface of a liquid (or solid) only has an indirect effect on the amount
of vapor present. The total pressure P on the liquid is now the sum
of the partial pressures P f of the foreign gas and P v of the vapor.
An addition of enough more of the foreign gas to increase this total
pressure by dP (keeping T constant) will increase the Gibbs func
tion of the liquid by dG x = V\ dP [see Eq. (811); dT = 0] , but the
Gibbs function of the same amount of material in gaseous form is not
affected by the foreign gas, so dGg = Vg dP v . For liquid and vapor to
remain in equilibrium dGj must equal dGg; consequently the relation
between vapor pressure P V (P,T) in the presence of a foreign gas and
the total pressure P is given by
dP v /dP = Vi/Vg
which may be integrated from the initial state where no foreign gas
is present [P = P v and P v is the solution of Eq. (96)] to the final
state where P = Pf + P v . Since Vg is so much larger than Vi, P v
changes very little as the foreign gas is added, but what change there
is, is positive. Addition of foreign gas squeezes a little more vapor
out of the liquid, rather than pushing some vapor back into the liquid.
Water in a dish open to the air is not in equilibrium unless the par
tial pressure of water vapor in the air happens to be exactly equal to
the vapor pressure P V (T) for the common temperature T of air and
water (this is the condition of 100 per cent humidity). If the common
temperature is above this, the water continues to evaporate until it is
all gone, the evaporation proceeding more rapidly as the temperature
is raised, until the boiling point is reached, when P V (T) is equal to
the total atmospheric pressure; the gas immediately above the water
is all water vapor, and the water boils rapidly away.
The latent heats of evaporation are usually 10 to 50 times greater
than the corresponding latent heats of melting, corresponding to the
fact that it takes much more work to pull the material into a tenuous
vapor than it does to change it into a liquid, which disrupts the crys
tal structure but doesn't pull the atoms much further apart.
74
THERMODYNAMICS
At very high pressures there are also phase changes in the solid
state; the crystal structure of the solid changes, with accompanying
latent heat, change of volume, and relationship between P and T for
the change given by an equation such as (96).
Triple Point and Critical Point
We have just seen that the melting temperature is nearly independ
ent of pressure, whereas the temperature of vaporization is strongly
dependent on P. Therefore as P decreases, the two curves, one for
T v , the other for T m , converge. This is shown in Fig. 93, where the
v
, 1 1
/
*°# 1
t
' s
^/ 1
' s
B
1&$
II 1 VI
^ 1 <^
£/ A/
/ 7l
7
*7
/ c /
/ >
/ l
/ /
/ /
i/^">
1 '
yt''
' '
m*
t / 1
/ /
1 y^
FIG. 93. Phase diagram for a material that expands upon
melting. Solid lines are the curves for phase
change, dashed lines those for constant volume.
curve AB is the meltingpoint curve and AC that for vaporization.
The two meet at the triple point A, which is the only point where solid,
liquid, and vapor can coexist in equilibrium. Below this pressure the
liquid is not a stable phase and along the curve OA the solid trans
forms directly into the vapor (sublimation). The shape of curve OA is
governed by an equation similar to (96), with a latent heat of sublima
tion L s (equal to L m + Ly at the triple point). The dashed lines of
Fig. 93 are lines of P against T for different values of V, intersec
tions of the PVT surface by planes parallel to the PT plane.
CHANGES OF PHASE
75
As the pressure is increased, keeping T = T V (P) so that we follow
the curve AC, the differences (VgV"i) and (S g Si) = (Ly/T v ) be
tween gas and liquid diminish until at C, the critical point, there
ceases to be any distinction between liquid and gas and the curve AC
terminates. There seems to be no such termination of the curve AB
for melting; the difference between the regularly structured solid and
the irregular liquid remains for pressures up to the maximum so far
attained; it may be that curve AB continues to infinity.
The PT plane is only one way of viewing the PVT surface repre
senting the equation of state. Another sometimes more useful projec
tion is the one on the PV plane. In Fig. 94 are plotted the dashed
curves of P against V for different values of T, corresponding to
Ml
l B tj
\ \
III
■ — ■
\ \
III
\ \
II I
II I
II \
II \
II \
\ * \ V\
II v
11 \
11 ]
3
[ 1 >°^ *V
1 ra
\ •s'/cV.. c ^
1 "
3
1 "V \ •» ■>. K ^
1 » l
1 » l
1 » l
o
CO
3
V \ x *
1 1 1
'3
L _ — — V n„
1 » l
1 » !
"1/ \> " ^
I 1 »
\ l 1
1/ \ v *«^. •*■
1/ vaporliquid \ ^<.
1 \ ^
\ V
A A ^ST Tj=jT t
\ / ~^^? ^ ,_
\/ vaporsolid ^*****^J" — —
FIG. 94. PV curves (dashed lines) for the material of
Fig. 93. Solid lines are projections on the PV
plane of the solid lines of Fig. 93.
intersections of the PVT surface with planes of constant T, parallel
to the PT plane. The regions where the PV curves are horizontal
are where there is a phase change. The boundaries of these regions,
ABA and ACA, projected on the PT plane, are the curves AB and
76
THERMODYNAMICS
AC of Fig. 93. It is clearer from Fig. 94, why C is the critical
point. The line AAA corresponds to the triple point A of Fig. 93
The SPT surface is also divided into the various phase regions.
Figure 95 shows a part of this surface, projected on the ST plane
the surface being ruled with the dashed lines of constant pressure.
v ' / '
•
1 N. /q, vapor
•
s
s
s
! A
/
^*\^ /
/
/
/
/
/ /
vaporsolid
vaporliquid
F gas
1 >7
/
1 ^ / ^ /
/
/ ,*<>''
A
solliq ■ i
i B
^c"— ""
solid
FIG. 95. Entropy as a function of T (dashed curves) for
various values of P, for the material of Fig. 93
We see that, as pressure is kept constant and T is increased, S in
creases steadily until a phase change occurs, when S takes a sudden
jump, of amount L/T, and then continues its steady increase with
temperature in the new phase. Entropy change is largest between solid
and vapor, not because the latent heat is so much larger for this phase
change, but because it occurs at low temperature and S<
Le/T s , where
T s is small.
>g
The GPT surface for the Gibbs function, projected on the GT
plane, is shown in Fig. 96. The dashed lines correspond to the inter
sections of the surface with planes of constant P, parallel to the GT
CHANGES OF PHASE
77
FIG. 96. Gibbs function versus temperature (dashed lines)
for various values of P, for the material of Fig.
93.
plane. The solid lines correspond to the phase changes. As noted ear
lier in this chapter, G does not change suddenly during a phase change,
as do V and S; only the slopes (9G/3P) T = V and (8G/3T) V  S (the
slopes of the dashed lines of Fig. 96) change dis continuously across
the phase change boundaries. By taking gradients the curves of Figs.
94 and 95 can be obtained from Fig. 96 or, vice versa, the curves
of Fig. 96 can be obtained by integration of the data on curves in
Figs. 94 and 95.
Chemical
Reactions
The Gibbs function also is of importance in describing chemical
processes. Since most chemical reactions take place at constant tem
perature and pressure, the reaction must go in the direction of de
creasing G.
Chemical Equations
A chemical reaction is usually described by an equation, such as
2H 2 + 2 — ■ 2H 2 0, which we can generalize as
Z>iMi^£>J M J
i J
stating that a certain number, i^, of molecules of the initial reac
tants Mi will combine to produce a certain number, v* 9 of molecules
of final products Mj. Of course the chemical reaction can run in either
direction, depending on the circumstances; we have to pick a direction
to call positive. This is arbitrarily chosen to be the direction in which
the reaction generates heat. We then, also arbitrarily, place all the
terms in the equation on the right, so that it reads
o = y>i M i (loi)
i
In this form the ^'s which are negative represent initial reactants and
those which are positive represent final products {v for the H 2 in the
example would be 2, v for the 2 would be 1, and v for the H 2 would
be +2). The number v\ is called the stoichiometric coefficient for M^
in the reaction.
For a chemical reaction to take place, more than one sort of mate
rial must be present either initially or finally, and the numbers n^ of
moles of the reacting materials will change during the reaction. Re
ferring to Eqs. (811) and (817) we see that the Gibbs function and its
change during the reaction are
78
CHEMICAL REACTIONS 79
G=E n iMi; dG = S dT + VdP+ J) i^i dn i (10~ 2 )
i i
where ix\ is the chemical potential of the ith component of the reac
tion, which we see is the Gibbs function per mole of the material M^
We shall see shortly how it depends on T and P and how it can be
measured.
A chemical reaction of the sort described by Eq. (101) produces a
change in the n's in an interrelated way. If dn x = p 1 dx moles of ma
terial M x appear during a given interval of time while the reaction
progresses, then dni = V{ dx moles of material Mi will appear during
the same interval of time (or will disappear, if v^ is negative). For
example, if 2dx moles of H 2 Q appear, simultaneously 2dx moles of
H 2 and dx moles of 2 will disappear (i.e., dnjj = 2dx and dn = dx)
In other words, during a chemical reaction at constant T and P, the
change in the Gibbs function is
dG=£ jut^idx
i
In accordance with the discussion of Eq. (811) we should expect the
reaction at constant T and P to continue spontaneously, with conse
quent decrease of G, until G becomes minimum at equilibrium. Thus,
at equilibrium, at constant T and P, dG/dx = 0, or
2 Mi^i = for equilibrium (103)
i
At equilibrium the relative proportions of the reactants and the prod
ucts must adjust themselves so that the /i's, which are functions of T,
P, and the concentration of the ith component, satisfy Eq. (103).
Heat Evolved by the Reaction
During the progress of the reaction, if carried out at constant T
and P, any evolution of heat would be measured as a change in en
thalpy since, as we have already remarked [see the discussion after
Eq. (87)] , enthalpy is the heat content of the system at constant pres
sure. But since, from Eq. (817),
H = G + TS = G TOG/9T) (104)
the change in H as the parameter x changes at constant T and P is
dH = (f h  x WHI)] pn * = t s ^  T (l^'i] <*
80 THERMODYNAMICS
At equilibrium £ jii^i = and the reaction ceases. If we measure the
heat evolved per change dx when the system is close to equilibrium,
the rate of evolution of heat becomes
dH/dx=T^2>i I 'i) n (105)
Thus by measuring the rates of change with temperature, (3^i/3T)p n ,
of the chemical potentials of the substances involved, we can predict
the amount of heat evolved during the reaction. Or, vice versa, if we
can measure (or can compute, by quantum mechanics) the heat evolved
when v^ moles of substance Mj appear or disappear, by dissociating
into their constituent atoms or by reassociating the atoms into the
product molecules, we can predict the rate of change of the ju's with
temperature.
Reactions in Gases
To see how all this works out, we take the simple case of a mixture
at high enough temperature so that all the components are perfect
gases. The number of moles of the ith component is n^ and the total
number of moles in the mixture is n = En^. The relative proportions
of the different gases can be given in terms of their concentrations^
Xi = nj/n, so that ^Xi = 1> or they can be expressed in terms of their
partial pressures Pi = XiP Each mode of expression has its advan
tages.
For example, we can compute the Gibbs function per mole i±\ of
Mi, as a perfect gas, in terms of T and pj. Using the procedures of
Chapter 8 [see Eqs. (821) for example] , we have, for a perfect gas
for which C p (T) = C V (T) + nR,
(aS/3T) p = (l/T)C p (T); (9S/9P) T = OV/3T) p = (nR/P)
so
T
S = S + / C p (T)(dT/T)  nR ln(P/P )
T
and, since
(9G/8T) = S and (3G/3P) T = V = nRT/P
we have
aG/ani = /ii = RT ln(Pi/P ) + gi (T) = RT[ln(P/P ) + In Xl ] + gi (T)
(106)
CHEMICAL REACTIONS
81
where
gi (T) = gi (T )s (TT )
T T
/ dT / c p (T)(dT/T)
s being the entropy per mole of the ith component at T = T and Cp
being the specific heat, the heat capacity per mole at constant pres
sure of Mi in its gaseous form.
The equation for chemical equilibrium (103) then takes on the form
RT£>i
i
:■(*)
+ In Xi +
gi
RT
]
or
v
nxi
1 = (Po/P)^ 1
K(T); K(T) = exp [£ ^igi(T)/RT] (10 7)
v\
where the sign n indicates the product of all the terms Xi for all
values of i.
The quantity K(T) is called the equilibrium constant and the equa
tion determining the x's at equilibrium is called the law of mass ac
tion. To see how this goes, we return to the reaction between H 2 and
2 . Suppose initially n x moles of H 2
and n 2 moles of 2 were present
and suppose during the reaction x
combined to form 2x moles of
present would be (n x  2x) + (n 2
moles of O and 2x moles of H,
H 2 0.
The total number of moles then
x) + 2x = n. + n 2  x, so the con
centrations at equilibrium and the stoichiometric coefficients for the
reaction are
for
H 2 :
X x = [(n 1 2x)/(n 1 + n 2 x)]
^=2
for 2 : x 2 = [(n 2 x)/(n 1 + n 2 x)]
1
for H 2 0: x 3 = [2x/(n 1 + n 2  x)]
^3 = 2
and the law of mass action becomes
2x(n 1 + n 2  x)
( ni 2x) 2 (n 2 x)
©
exp
2 gl + g 2 2g ;
RT
()k(t)
from which we can solve for x. Since this particular reaction is
strongly exothermic, 2g x + g 2 is considerably larger than 2g 3 and the
exponential K(T) is a very large quantity unless T is large. Conse
quently, for P « P and for moderate temperatures, x will be close
to (l/2)n x or ng, whichever is smaller, i.e., the reaction will go al
most to completion, using up nearly all the constituent in shorter sup
ply. If there is a deficiency in hydrogen, for example, so that n 1 < 2IL,,
then we set x = (l/2)n x  6, where 6 is small, and
82
THERMODYNAMICS
(26) " [D,(l/2)nJ \P^ exp
Xi~
26
1 ^+(1/2)^'
RT
_ n 2 ~ (1/2)^ n t
X2 "n 2 +(l/2) ni ' X 3" ^+(1/2)11,
the system coming to equilibrium with a very small amount, 26 moles
of H 2 left. This amount increases with decrease of pressure P and
with increase of T.
To see how the equilibrium constant K(T) changes with tempera
ture, we can utilize Eqs. (103), (105), and (106). We have
_d_
dT
In K =
1
RT 2
E^igi T X>iSi
= ^[ T (^HJ^ 2 (i) TP (m
where g^ = (dgj/dT). The rate of change of the equilibrium constant
K(T) with temperature is thus (for gas reactions) proportional to the
amount of heat evolved (5H/9x)pT per unit amount of reaction at
equilibrium at constant T and P. This useful relationship is known
as Van't Hoff's equation.
Electrochemical Processes
Some chemical reactions can occur spontaneously in solution, gen
erating heat, or they can be arranged to produce electrical energy in
stead of heat. For example , a solution of copper sulfate contains free
copper ions in solution. The addition of a small amount, An moles, of
metallic zinc in powder form will cause An moles of copper to appear
in metallic form, the zinc going into solution, replacing the Cu ions.
At the same time an amount WAn of heat is released. If the reaction
takes place at constant pressure and temperature, Eq. (87) indicates
that W An must equal the difference in enthalpy between the initial
state, with Cu in solution, and the final state, with Zn in solution,
Hjl  H 2 = WAn
(109)
However the same reaction can take place in a battery, having one
electrode of copper in a CuS0 4 solution and the other electrode (the
negative pole of the battery) of zinc, surrounded by a ZnS0 4 solution,
CHEMICAL REACTIONS 83
the two solutions being in contact electrically and thermally. If the
battery now discharges an amount AC coulombs of charge through a
resistor, or a motor to produce work, more Cu will be deposited on
the Cu electrode and an equal number of moles, An, of Zn will leave
the zinc electrode to go into solution. In this case the energy of the
reaction goes into electromechanical work; for every charge AC dis
charged by the battery, 8 AC joules of work are produced, where 8
is the equilibrium voltage difference between the battery electrodes.
If the battery is kept at constant temperature and pressure during
the quasistatic production of electrical work, Eq. (8ll)(which in this
case can be written dG = S dT + V dP + 8 dC) shows that the work
done equals the change in the Gibbs function caused by the reaction.
In other words 8 AC = AG. But Eq. (104) shows the relationship be
tween enthalpy and Gibbs function, which in this case can be written
G = H x  H 2 + T(3 AG/3T) p (10 10)
which, with Eq. (109), provides a relationship between the electrical
properties of the battery and the thermal properties of the related
chemical reaction.
If the ions have a valency z (for Zn and Cu, z = 2) then a mole of
ions possesses a charge z £F, where 5 is the Faraday constant 9.65 x
10 7 coulombs per mole. Thus the charge AC is equal to zJFAn, where
An is the number of moles of Zn that goes into solution (or the number
of moles of Cu that is deposited). Combining Eqs. (109) and (1010),
we obtain the equation (using AG = zJ8 An and dividing by z^ An)
8 = (W/zff) + TOs/9T) p (1011)
relating the emf of the cell and the change in emf with temperature to
the heat W evolved in the corresponding chemical reaction; and, by
use of Eq. (105), we have derived a means of obtaining empirical
values of the chemical potentials /ij. Thus electrical measurements,
which can be made quite accurately, can be used instead of thermal
measurements to measure heats of reaction. Equation (1011) is
called the Gibbs Helmholtz equation.
IT
KINETIC THEORY
Probability and
Distribution
Functions
We have now sketched out the main logical features of thermody
namics and have discussed a few of its applications. We could easily
devote the rest of this text to other applications, covering all branches
of science and engineering. But, as physicists, it is more appropriate
for us to go on to investigate the connection between the thermal prop
erties of matter in bulk and the detailed properties of the atoms that
constitute this matter. The connection, as we saw in Chapter 2, must
be a statistical one and thus will be expressed in terms of probabili
ties.
Probability
A basic concept is hard to define, except circularly. Probability is
in part subjective, a quantization of our expectation of the outcome of
some event (or trial) and only measurable if the event or trial can be
repeated several times. Suppose one of the possible outcomes of a
trial is A. We say that the probability that the trial results in A is
P(A) if we expect that, out of series of N similar trials, roughly
NP(A) of them will result in A. We expect that the fraction of the
trials which do result in A will approach P(A) as the number of trials
increases. A standard example is the gambler's sixsided die; a 5
doesn't come up regularly every sixth time the die is thrown, but if a
5 comes up 23 times in 60 throws and also 65 times in the following
240 throws, we begin seriously to doubt the symmetry of the die
and/or the honesty of the thrower.
If result A occurs for every trial, then P(A) = 1. If other events
sometimes occur, such as event B, then the probability that A does
not occur in a trial, 1  P(A), is not zero. It may be possible that
both A and B can occur in a single trial; the probability of this hap
pening is written P(AB). Simple logic will show that the probability of
either A or B or both occurring in a trial is
87
88 KINETIC THEORY
P(A + B) = P(A) + P(B)  P(AB) (111)
Relationships between probabilities are often expressed in terms of
the conditional probabilities P(AB) that A occurs in a trial if B
also occurs, and P(BA) the probability that B occurs if A occurs
as well. We can see that
P(AB) = P(AB) P(B) = P(BA) P(A) = P(BA) (112)
A simple example is in the dealing of well shuffled cards. The
chance of a heart being dealt is P(H) = 1/4, the chance that the card
is a seven if it is a heart is P(7H) = 1/13 and therefore the proba
bility that the card dealt is the seven of hearts is P(7H) = P(7H) P(H)
= (l/13)(l/4) = 1/52.
If the probability of A occurring in a trial is not influenced by the
simultaneous presence or absence of B, i.e., if P(AB)  P(A), then
A and B are said to be independent. When this is the case,
P(AB) = P(A); P(BA) = P(B); P(AB) = P(A) P(B)
[1  P(A + B)] = [1  P(A)][1  P(B)] (113)
Saying it in words, if A and B are independent, then the chance of
both A and B occurring in a trial is the product P(A) P(B) of their
separate probabilities of occurrence and the probability 1  P(A + B)
that neither A nor B occur is the product of their separate probabil
ities of nonoccurrence. In the example of the dealing of cards, since
the chance P(7) of a seven being dealt is the same as the conditional
probability P(7H) that a seven is dealt if it is a heart, the occurrence
of a seven is independent of the occurrence of a heart. Thus the prob
ability that the card dealt is a seven of hearts is P(7) P(H) = (l/13)(l/4)
= 1/52, and the chance that it is neither a seven nor a heart is
(12/13)(3/4) = 9/13.
If the trial is such that, when A occurs B cannot occur and vice
versa, then A and B are said to be exclusive and
P(AB) = P(AB) = P(BA) =
so that
P(A + B) = P(A) + P(B) (114)
The chance of either A or B occurring, when A and B are exclusive,
is thus the sum of their separate probabilities of occurrence. For ex
ample, the result that a thrown die comes up a 5 is exclusive of its
PROBABILITY AND DISTRIBUTION FUNCTIONS 89
coming up a 1; therefore the chance of either 1 or 5 coming up is
(1/6) + (1/6) =1/3.
Probabilities are useful in discussing random events. The defini
tion of randomness is as roundabout as the definition of probability.
The results of successive trials are randomly distributed if there is
no pattern in the successive outcomes, if the only prediction we can
make about the outcome of the next trial is to state the probabilities
of the various outcomes.
Binomial Distribution
The enumeration of the probabilities of all the possible outcomes
of an event is called a distribution function, or a probability distribu
tion. For example, suppose the ''event" consists of N independent
and equivalent trials, such as the throwing of N dice (or the throwing
of one die N times). If the probability of "success" in one trial is p
(if "success" is for the die to come up a 1, for example, then p = 1/6),
it is not difficult to show that the probability of exactly n successes in
N trials (not distinguishing the order of failures and successes) is
Pn(N) = n!(N N i n) , P n (lp) N  n (H5)
All possible results of the event are included in the set of values of n
from to N, so the set of probabilities P n (N) is a distribution func
tion; this particular one is called the binomial distribution.
When the possible results of the event are denumerable, as they
are for the binomial distribution, the individual probabilities can be
written P n , where n is the integer labeling one of the possible re
sults. If the results are mutually exclusive,
£P n =l (H6)
n
where the sum is taken over the values of n corresponding to all pos
sible results. For example, for the binomial distribution,
P n (N) =J nt(Nn)l p " (1 ~ p) N_ " = k + U p)] N = 1
Suppose the value of result n is x(n). The expected value <x> of
an event is then the weighted average
<x> = £x(n) P n (117)
n
90 KINETIC THEORY
Any individual event would not necessarily result in this value, but we
should expect that the average value of a series of similar events
would tend to approach <x> . A measure of the variability of individ
ual events is the variance (o" x ) 2 ot x, the mean square of the differ
ence between the actual value of an event and its expected value,
(a x ) 2 = J[x(n)  <x>] 2 P n = <x 2 >  <x> 2 (118)
n
The square root of the variance cr x is called the standard deviation of
x for the particular distribution.
Random Walk
To make these generalities more specific let us consider the proc
ess called the random walk. We imagine a particle in irregular motion
along a line; at the end of each period of time t it either has moved a
distance 5 to the right or a distance 6 to the left of where it was at
the beginning of the period. Suppose the direction of each successive
"step" is independent of the direction of the previous one, that the
probability that the step is in the positive direction is p and the prob
ability that it is in the negative direction is 1  p. Then the probability
that during N periods the particle has made n positive steps, and
thus Nn negative steps, is the binomial probability P n (N) of Eq.
(115).
The net displacement after these N periods, x(n) = (2nN)6, might
be called the "value" of the N random steps of the particle. The ex
pected value of this displacement after N steps can be computed in
terms of the expected value <n> of n,
<n>=f;nP n (N) = p I ^ 1 k , (N _ N 1 ! . k) , pk(lp) N  1  k
n=0 k=0
pN
so
<x(n)>= <(2nN)6>= (2pl)NS; k = nl (119)
When p = 1 (the particle always moves to the right) the expected dis
placement after N steps is N6; when p = (the particle always
moves to the left) it is N6; when p = 1/2 (the particle is equally
likely to move right as left at each step) the expected displacement is
zero. The variability of the actual result of any particular set of N
steps can be obtained from <n> and <n 2 >,
<n 2 >= <n(n l)> + <n>= p 2 N(N 1) + pN
PROBABILITY AND DISTRIBUTION FUNCTIONS 91
so that
(<j x ) 2 = < [x(n)  N(2p  l)6] 2 > = < (2n  2pN) 2 5 2 >
= 4Np(l  p)6 2 (1110)
When p = 1 (all steps to right) or p =0 (all steps to left) the variance
(a x ) 2 of the displacement is zero; when p = 1/2 the variance is great
est, being N6 2 (the standard deviation is then cr x = 6 VN).
The Poisson Distribution
When the outcome of an event has a continuous range of values, the
sums discussed heretofore must be changed to integrals and we must
talk about a probability density f(x), defined so that f(x) dx is the
probability that the result lies between x and x + dx. The representa
tive example is that of dots distributed at random along a line. The
line may be the distance traveled by a molecule of a gas and the dots
may represent collisions with other molecules, or the line may be the
time axis and the dots may be the instants of emission of a gamma
ray from a piece of radioactive material; in any of these cases a dot
is equally likely to be found in any element dx of length of the line.
If the mean density of dots is (1/x) per unit length (A being thus a
mean distance between dots) then the chance that a randomly chosen
element dx of the line will contain a dot is (dx/A), independent of the
position of dx.
We can now compute the probability P (x/A) that no dot will be
found in an interval of length x of the line, by setting up a differential
equation for P . The probability P [(x + dx)/A] that no dots are in an
interval of length x+dx is equal to the product of the probability
P (x), that no dots are in length x, times the probability 1  (dx/A),
that no dot is in the additional length dx [see Eq. (113)]. Using
Taylor's theorem we have
■>.(^).(f)^.(f)*='.g)['(?)
or
£*6)(iW!) »* p og)=e*A uiii)
The probability that no dot is included in an interval of length x,
placed anywhere along the line, thus decreases exponentially as x in
creases, being unity for x = (it is certain that no dot is included in
92 KINETIC THEORY
a zero length interval), being 0.368 for x = A, the mean distance be
tween dots (i.e., a bit more than a third of the intervals between dots
are larger than A) and dropping off rapidly to zero for x larger
than A.
The probability density for this distribution is the derivative of P ;
it supplies the answer to the following question. We start at an arbi
trary point and move along the line: What is the probability that the
first dot encountered lies between x and x + dx from the start? Equa
tion (113) indicates that this probability, f(x) dx, is equal to the prod
uct of the probability e~ x /^ that no dot is encountered in length x,
times the probability dx/A that a dot is encountered in the next inter
val dx. Therefore
f(x) = (1/A)e" x/A (1112)
is the probability density for encountering the first dot at x (i.e., f dx
is the probability that the first dot is between x and x + dx).
As with discrete distributions, one can compute expected values and
variances for probability densities. For example, the expected dis
tance one goes, from an arbitrarily chosen point on the line, before a
dot is encountered is
OO 00
<x> = /xf(x) dx= A /u e~ u du = A (1113)
for the randomly distributed dots. The variance of this distance is
oo
(o x ) 2 = f (xA) 2 f(x)dx = A 2 (1114)
so the standard deviation of x, cr x = A, is as large as the mean value
A of x, an indication of the variability of the interval sizes between
the randomly placed dots.
We can go on to ask what is the probability P 1 (x/A) of finding just
one dot in an interval of length x of the line. This is obtained by us
ing Eq. (113) again to show that the probability of finding the first dot
between y and y + dy from the beginning and no other dot between this
and the end of the interval x is equal to the product f(y) dy P [(xy)/A]
Then we can use Eq. (114) to show that the probability P x (x/A) that
the one dot is somewhere within the interval x is the integral
Pl (x/A) = /(l/A)e y/A dy e (x " y)A = (x/A) e" x/A
An extension of this argument quickly shows that the probability that
there are exactly n dots in the interval x is
PROBABILITY AND DISTRIBUTION FUNCTIONS 93
P n (x/X) = /f(y) dy P n _ ^SZI) = &f eA
We have thus derived another discrete distribution function, called
the Poisson distribution, the set of probabilities P n (x/x), that n dots,
randomly placed along a line, occur in an interval of length x of this
line. More generally, a sequence of events has a Poisson distribution
if the outcome of an event is a positive integer and if the probability
that the outcome is the integer n is
oo
P n (E) = (E n /n!)e" E ; J^ P n (E) = 1 (1115)
n =
If E < 1, P (E) is larger than any other P n ; if E > 1, P n is maxi
mum for n ^E, tending toward zero for n larger or smaller than
this. Quantity E is the expected value of n, for
oo °o
< n > = J^ nP n (E) = E and (a n ) 2 =^T (n  E) 2 P n (E) = E
n = n = (n . 16)
Arrivals of customers to a store during some interval of time T or
the emission of particles from a radioactive source in a given inter
val of time— both of these have Poisson distributions. We shall en
counter distributions like these later on.
The Normal Distribution
In the limit of large values of N for the binomial distribution or of
large values of E for the Poisson distribution, the variable n can be
considered to be proportional to a continuous variable x, and the
probability P n approaches, in both cases, the same probability den
sity F(x). This function F(x) has a single maximum at x = <x> , the
expected value of x (which we shall write as X, for the time being,
to save space). It drops off symmetrically on both sides of this maxi
mum, the width of the peak being proportional to the standard devia
tion a, as shown in Fig. 111. Function F has the simplest form
compatible with these requirements, plus the additional requirement
that the integrals representing expected values converge,
F(x ~ X)= ^f e " (X_X)V2CT2; /F(xX)dx=l
— oo
oo
<x>= / xF(xX) dx = X; <(xX) 2 > = a 2 (1117)
94
KINETIC THEORY
FIG. 111. The normal distribution
This is known as the normal, or Gaussian, distribution. It is typi
cal of the behavior of a system subject to a large number of small,
independent random effects. As an example, we might take the limit
ing case of a symmetric random walk (p = 1/2), where the number N
of steps is large but the size of each step is small, say 6  cr/VN.
Then, if a particular sequence of steps turns out to have had n posi
tive steps and Nn negative ones, the net displacement would be
x = a (2n  N)/VN and the probability of such a displacement would be
p (N) = N!(1/2) N
nW [(1/2)N + (x/2a)VN]![(l/2)N  (x/2a)VNJ !
since n = (1/2)N + (x/2cr)VN.
When N is very large, x tends to become practically a continuous
variable and in the limit P n (N)dn^ F dx, where F is the probability
density
F = lim
N — oo
= lim
N— o
^ p (N)
2a nW
(VN/2a)N!(l/2) N
[(1/2 )N + (x/2a)VN]![(l/2)N (x/2a)VN]l
To evaluate this for large
factorial function,
N we use the asymptotic formula for the
n! — V27fn n n e~ n :
> 10
(1118)
PROBABILITY AND DISTRIBUTION FUNCTIONS
95
which is called Stirling's formula. Using it for each of the three fac
torials and rearranging factors, we obtain
F = lim
QyfZn Vl(x 2 /a 2 N) V aVN
1 +
x 1
—
aVN/
(1/2)N  (x/2a)VN
■(l/2)N + (x/2a)VN
lim
cjV^
o 2 N
(l/2)Nl/2
1 
aVN
xv^/2a
1 +
aVN
xVN/2a
By using the limiting definition of the exponential,
lim (l + )
n— °° \ n/
(1119)
this expression reduces to that of Eq. (1117), with X = 0. The termi
nal point of a random walk with a large number of small steps is dis
tributed normally, as is any other effect which is the result of a large
number of independent, random components. A proof that the limiting
form of the Poisson distribution also is normal constitutes one of the
problems. The distribution of random errors in a series of measure
ments is usually normal, as is the distribution of shots at a target. In
fact the normal distribution is wellnigh synonymous with the idea of
randomness.
Velocity
Distributions
Probability distributions are the connecting link between atomic
characteristics and thermodynamic processes. We mentioned in Chap
ter 1 that each thermodynamic state of a system corresponded to any
of a large number of microstates, macroscopically indistinguishable
but microscopically different configurations of the system's atoms. If
we had an assembly of thermodynamically equivalent systems, any
one of the systems may be in any one of this large set of microstates;
indeed each one of the systems will pass continuously from one micro
state to another of the set. All we can specify is the probability i\ of
finding the system (any one of them) in microstate i. In fact a speci
fication of the distribution function fj, specifying the value of fj for
each microstate possible to the system in the given macrostate, will
serve to specify the thermodynamic state of each of the systems of
the assembly.
Momentum Distribution for a Gas
In the case of a perfect gas of N point atoms, each atom is equally
likely to be anywhere within the volume V occupied by the gas, but the
distributioninvelocity of the atoms is less uniformly spread out.
What is needed is a probability density function, prescribing the prob
ability that an atom, chosen at random from those present, should have
a specified momentum, both in magnitude and direction. We can vis
ualize this by imagining a threedimensional momentum space, as
shown in Fig. 121, wherein the momentum p of any atom can be
given either in terms of its rectangular components p x , p y , p z or
else in terms of its magnitude p and its spherical direction angles a
and /3. The probability density f is then a function of p x , p y , p z or
of p, a, /3 (or simply of the vector p) such that f(p) dV p is the prob
ability that an atom of gas will turn out to have a momentum vector p
whose head lies within the volume element dV p in momentum space
(i.e., whose x component is between p x and p x + dp x , y component
96
VELOCITY DISTRIBUTIONS
97
Fig. 121. Coordinates in momentum space.
is between p y and p y + dp y , and z component is between p z and
Pz + d Pz> where dV p = dp x dp y dp z = p 2 dp sin a dot d/3).
We do not assume that any given atom keeps the same momentum
forever; indeed its momentum changes suddenly, from time to time,
as it collides with another atom or with the container walls. But we
do assume, for the present, that these collisions are rather rare
events and that if we should observe a particular atom the chances
are preponderantly in favor of finding it between collisions, moving
with a constant momentum, and that the probability that it has mo
mentum p is given by the distribution function f(p) which has just
been defined.
If the state of the gas is an equilibrium state we would expect that
f would be independent of time; if the state is not an equilibrium one,
f may depend on time as well as on p. By the basic definition of a
probability density, we must have that
oo oo oo
///f(p)dVp= /dp x Jdp v Jd Pz f(p)
277 77 oo
= /d/3 /sin a da /f(p)p 2 dp = l
(121)
since it is certain that a given atom must have some value of momen
tum. The distribution function will enable us to calculate all the vari
ous average values characteristic of the particular thermodynamic
state specified by our choice of f(p). For example, the mean kinetic
energy of the gas molecule (between collisions, of course) is
277 77 oo
<K.E.> tran = /d/3 Jsina da /f(p)(p 2 /2m) p 2 dp (122)
vwc3680 WAB Morse
3P
10
■2761
98 KINETIC THEORY
and the total energy of a gas of point atoms would be N< K.E. > ,
where N is the total number of atoms in the system [see Eq. (21)].
If the gas is moving as a whole, there will be a drift velocity V
superimposed on the random motion, so that f(p) is larger in one di
rection of p than in the opposite direction, more atoms going in the
positive x direction (for example) than in the negative x direction.
In this case the components of the drift velocity are
oo oo oo
V x = <p x /m> = J(p x /m)dp x jdp Jdp z f(p)
oo _oo J _0O
277 77 oo
= J d/3 /sin a da J(p cos a/m) f(p) p 2 dp (123)
and similarly for V v and V z .
If the gas is in equilibrium and its container is at rest the drift ve
locity must of course be zero. In fact at equilibrium it should be just
as likely to find an atom moving in one direction as in another. In
other words, for a gas at equilibrium in a container at rest we should
expect to find the distribution function independent of the direction an
gles a and /3 and dependent only on the magnitude p of the momen
tum. In this case V = and
4tt / f(p) p 2 dp = 1; <K.E.>=— /f(p)p 4 dp (124)
The Maxwell Distribution
We now proceed to obtain, by a rather heuristic argument, the mo
mentum distribution of a gas of point atoms in equilibrium at tempera
ture T. A more "fundamental" derivation will be given later; at pres
ent under standability is more important than rigor. We have already
seen that at equilibrium the distribution function should be a function
of the magnitude p of the momentum, independent of the angles a and
£. One additional fact can be brought to bear: Eq. (113) states that,
if the magnitudes of the three components of the momentum are dis
tributed independently, then f(p) should equal the product of the prob
ability densities of each component separately, f(p) =F(p x ) ' F(py) • F(p z ),
each of the factors being similar functions of the three components.
Moreover, since the atomic motions are entirely at random, it
would seem reasonable that the function F should have the form of
the normal distribution, Eq. (1117), which represents the effects of
randomness. Thus we would expect that the equilibrium momentum
distribution would be
f (p) = F(p x )F(p y )F(p z ) = j^g; exp  p2 *~Py 2 ~ P (12 .5)
where a is the standard deviation of either of the momentum compo
nents from its zero mean value. This result strengthens our impres
VELOCITY DISTRIBUTIONS 99
sion that we are on the right track, for the sum p^+ p^+ p = p 2 is
independent of the angles a and /3 , and thus f is a function of the
magnitude of p and independent of a and /3. In order to have f(p)
be, at the same time, the product of functions F of the individual
components and also a function of p alone, f must have the exponen
tial form of Eq. (125) (or, at least, this is the simplest function which
does so).
To find the value of the variance cr 2 in terms of the temperature of
the gas, we have recourse to the results of Chapters 2 and 3, in par
ticular of Eq. (32), relating the mean kinetic energy of trans lational
motion of the atoms in a perfect gas with the temperature T. To com
pute mean values for the normal distribution we write down the fol
lowing integral formulas:
1° u 2 /a . 1 _
J e du =  v^ra
o l
T u 2 /a 2n , 113 5 2n 1 n /i0 c ,
J e ' u du=2^^2 , 2'2'" — 2~ a (126)
u 2 /a 2n+l , _ 1 , n+1
r uya znti , i , i
J e u du =  n ! a
Therefore the mean value of the atomic kinetic energy is
00 °°
<K.E.> = 477 / (p 2 /2m)f(p)p 2 dp  f— J p 4 e~ p/2a dp
o mo" i v2rf o
=  (a 2 /m)
which must equal (3/2)kT, according to Eq. (32). Therefore the vari
ance a 2 is equal to mkT, where m is the atomic mass, k is Boltz
mann's constant, and T is the thermodynamic temperature of the gas,
Thus a rather heuristic argument has led us to the following mo
mentum distribution for the translational motion of atoms in a perfect
gas at temperature T,
f(p) = (27rmkT) 3 /2 e P 2 /2mkT (l2 _ 7)
which is called the Maxwell distribution. It is often expressed in
terms of velocity v = p/m instead of momentum. Experimentally we
find that it corresponds closely to the velocity distribution of mole
cules in actual gases. The distribution is a simple one, being iso
tropic, with a maximum at p = and going to zero at p~ °° . The in
tegral giving the fraction of particles that have speeds larger than v
is
100 KINETIC THEORY
a f *t \ 2 j 2 mv 2 /2kT /■
4ti J f (p)p 2 dp = = e J
mv ^ o
e
2 i 1/2
mv
2kT +U
du
^1.131^v72kTe mv2/2kT
when mv 2 > 2kT
Thus about half of the atoms have speeds greater than V2kT/m, about
1/25 of them have speeds greater than 2V2kT/m, and only one atom
in about 2400 has a speed greater than 3V2kT/m.
Mean Values
The mean velocity of the atoms is, of course, zero, since the mo
mentum distribution is symmetric. The mean speed and the mean
squared speed are
/ \ / / \ 47r f , ~P 2 /2mkT dp ,/8kT
<v> = <p/m>= — J p 3 e i// jz ,,v 3/2 = /
tf/ m o (27rmkT) 3 / 2 V 7im
<v 2 > = <p 2 /m 2 > =  (kT/m) (128)
We note that the mean of the square of the speed is not exactly equal
to the square of the mean speed (8/tt is not exactly equal to 3/2, al
though the difference is not large). The mean molecular kinetic en
ergy of translation is proportional to T; the mean molecular speed is
proportional to VT.
If the gas is a mixture of two kinds of molecules, one with mass m lt
the other with mass m 2 , then each species of molecule will have its
own distribution in velocity, the one with m l , instead of m in the ex
pression of Eq. (127), the other with m 2 instead of m. This is equiv
alent to saying that the mean kinetic energy of translational motion of
each species is (3/2)kT, no matter what the molecular weight of each
is, as long as the two kinds are in equilibrium at temperature T. In
fact if a dust particle of mass M is floating in the gas, being in equi
librium with the molecules of the gas, it will be in continuous, irreg
ular motion (called Brownian motion), which is equivalent to the ther
mal motion of the molecules, so its mean kinetic energy of translation
will also be (3/2)kT. Its meansquare speed, of course, will be less
than the meansquare speed of a gas molecule, by a factor equal to the
square root of the ratio of the mass of the molecule to the mass of the
dust particle.
Finally, we should check to make sure that a gas with molecules
having a Maxwell distribution of momentum will have a pressure cor
responding to the perfect gas law of Eq. (31). Of those molecules
which have an x component of momentum equal to p x , (N/V) dA (p x /m)
of them will strike per second on an area dA, perpendicular to the x
axis, N/V being the number per unit volume and (p x /m)dA being the
volume of the prism, in front of dA, which contains all the molecules
VELOCITY DISTRIBUTIONS 101
that will strike dA in a second. Each such molecule, striking dA,
would impart a momentum 2p x if dA were a part of the container
wall, so that the average momentum given to dA per second, which is
equal to the pressure times dA, is (see Fig. 21)
oo oo °o
P dA = (N/V) dA / dp z / d Py / (2p 2 x /m) f(p) dp x
 oo  oo
or
*  i^w I e " p2x/2mkT (2p * /m) dp *i ^" (p2y+ P?z)/2mkT
x dp y dp z
 I V3^ i * e"Px/ 2mkT d Px = NkT/V = nRT/V
(129)
where integration is only for positive values of p x since only those
with positive values of p x are going to hit the area dA in the next
second; the ones with negative values have already hit. Thus a gas
with a Maxwellian distribution of momentum obeys the perfect gas
law.
Collisions between Gas Molecules
Most of the time a gas molecule is moving freely, unaffected by the
presence of other molecules of the gas. Occasionally, of course, two
molecules collide, bouncing off with changed velocities. Roughly speak
ing, if two molecules come within a certain distance R of each other
their relative motion is affected and we say they have collided; if their
centers are farther apart than R they are not affected. To each mole
cule, all other molecules behave like targets, each of area a c = 7rR 2 ,
perpendicular to the path of the molecule in question. If the path of
this molecule's center of mass happens to intersect one of these tar
gets, a collision has occurred and the path changes direction. Since
there are N/V molecules in a unit volume, then in a disk like re
gion, unit area wide and dx thick, there are (N/V) dx molecules.
Therefore, the fraction of the disk obstructed by targets is (Na c /V) dx
and consequently the chance of the molecule in question having a col
lision while moving a distance dx is (Na c /V) dx. Target area cr c is
called the collision cross section of the molecules.
Thus a collision comes at random to a molecule as it moves
around; the density (1/a) of their occurrence along the path of its mo
tion is Na c /V. Reference back to the discussion before and after Eqs.
(1111) and (1112) indicates that if the chance of encountering a
"dot" (i.e., a collision) is dx/A, A is the mean distance between col
102 KINETIC THEORY
lisions (or dots) and the probability that the molecule travels a dis
tance x without colliding and then has its next collision in the next dx
of travel is
f(x) dx = (dx/A) e" x / A where A = (V/a c N) (1210)
The mean distance between collisions A is called the mean free path
of the molecule. We see that it is inversely proportional to the den
sity (N/V) of molecules and also inversely proportional to the molec
ular cross section a c . This mean free path is usually considerably
longer than the mean distance between molecules in a gas. For exam
ple, a c for 2 molecules is roughly 4x 10" 19 m 2 and N/V at stand
ard pressure and temperature (0° C and 1 atm) is approximately 2.5
x 10 25 molecules per m 3 . The mean distance between molecules is
then the reciprocal of the cube root of this, or approximately 3.5
x 10" 9 m, whereas the mean free path is A= V/Na c en 10" 7 m, roughly
30 times larger. The reason, of course, is that the collision radius R
is about 4 x 10" 10 m, for 2 , about one tenth of the mean distance be
tween molecules; thus only about 1/1000 of the volume is ' 'occupied"
by molecules at standard conditions. The difference is even more
marked at low pressures. At 10~ 7 atmosphere A is 1 meter in length,
but there are still 2.5 x 10 18 molecules per m 3 at this pressure, hence
the mean distance between molecules is roughly 0.7 x 10~ 6 m. Neither
A nor (7 C is dependent on the velocity distribution of the molecules.
We can also talk about a mean time r between collisions, although
this quantity does depend on the molecular velocity distribution. For
a molecule having a speed p/m it would take, on the average, mA/p
seconds for it to go from one collision to the next. Therefore the
mean free time r for a gas with a Maxwellian distribution of momen
tum is
r  < m */ P > = 4,nu / p **/*»« ^^ = X J/J
(1211)
which is not exactly equal to either A/<v> or A//<v 2 > but it is not
very different from either. The mean free time decreases with in
creasing temperature because an increase in temperature increases
the mean molecular speed, whereas it does not change A. Since the
mean speed of an oxygen molecule at standard conditions is about
400 m/sec, the mean free time for these conditions is about 3x 10" 10
sec; at 0°C and 10" 7 atm it is about (1/300) sec.
The Maxwell
Boltzmann
Distributions
When the mean density of particles in a gas is not uniform through
out the gas, or when electric or gravitational forces act on the mole
cules, then the distribution function for the molecules depends on po
sition as well as momentum, and may also depend on time. In such
cases we must talk about the probability f(r,p,t) dV r dV p that a mol
ecule is in the position volume element dV r = dx dy dz at the point
x, y, z (denoted by the vector r) within the container, and has a mo
mentum in the momentum volume element dV p = dp x dpy dp z at p x ,
Py> Pz (denoted by the vector p) at time t. Because f is a probability
density we must have /dV r JdV p f(r,p,t) = 1 , where the integration
over dV r is over the interior of the container enclosing the gas and
where the integration over dVp is usually over all magnitudes and di
rections of p.
Phase Space
To determine the dependence of f on r, p, and t we shall work out
a differential equation which it must satisfy. The equation simply
takes into account the interrelation between the force on a particle,
its momentum, and its position in space. In addition to the time vari
able , f is a function of six coordinates, the three position coordinates
and the three momentum coordinates. A point in this six dimensional
phase space represents both the position and momentum of the parti
cle. As the particle moves about in phase space, its momentum coor
dinates change in accordance with the force on the particle and its po
sition coordinates change in accordance with the momentum it has.
Each molecule of the gas has its representative point in phase space;
all the points move about as a swarm, their density at any time and
any point in phase space being proportional to f(r,p,t).
The point which is at (x,y,z,p x ,p y ,p z ) in phase space has a six
dimensional "velocity" which has components (x,y,z,p x ,py,p z ), where
the dot represents the time differential. We note the fact that the
103
104 KINETIC THEORY
' 'coordinates" are related to the "velocity" components in a rather
crosswise manner, for the x,y,z part of the velocity, f = p/m, is
proportional to the momentum part p of the phasespace coordinates,
independent of r; and the momentum part of the velocity, p = F(r), is
the particle's rate of change of momentum, which is equal to the force
F on the particle, which is a function of r. Thus two points in phase
space which have the same space components x,y,z but different mo
mentum coordinates p x ,Py>Pz have "velocities" with equal momen
tum components p x ,p v ,p z [because F(r) is the same for both points]
but different space components x,y,z (since the p's differ for the two
points). Vice versa, two points which have the same momentum co
ordinates but different space coordinates will have six dimensional
velocities with the same space components x,y,z but different momen
tum components p x ,p v ,p z .
At the instant of collision, the two points in phase space, represent
ing the two colliding molecules, suddenly change their momentum co
ordinates. In other words, those two points in phase space disappear
from their original positions and reappear with a new pair of momen
tum coordinates. Any text in hydrodynamics will show that the expres
sion (dp/dt) + div pv, where p is the fluid density and v its velocity,
is equal to the net creation of fluid at the point x,y,z. Extension to
phase space indicates that
ay dz ap x Njr * ap y y 3p z
at ' ' ' ■' ~ p
£ +div r (rf) + divjpf ) (131)
measures the net appearance of points in a sixdimensional volume
element at (x,y,z,p x ,p v ,p z ), the difference between collisionproduced
appearances and disappearances per second. Thus the expression
(131) should equal a function Q(r,p,t) called the collision function,
the form of which is determined by the nature of the molecular colli
sions.
The Boltzmann Equation
As stated two paragraphs ago, the "velocity" components x,y,z
are independent of r and the components p x , py, p z are independent
of the p's (but depend on r). Therefore Eq. (131) becomes
af , .. ■ af • . 8f , . af .. af , . af , . af ^
7
at ax J ay az ^ A ap x ^y ap v ^ ap
or
THE MAXWELLBOLTZMANN DISTRIBUTION 105
Of/at) + (p/m) • grad r f + F grad p f = Q(r,p,t) (132)
where F(r) is the force on a gas molecule when it is at position r.
This equation, which is called the Boltzmann equation, can also be ob
tained by assuming that the swarm of points, representing the mole
cules, as it moves along in phase space, only changes its density be
cause of the net difference between appearances and disappearances
produced by collisions.
The density of the swarm at r,p,t is proportional to f(r,p,t). A
time dt later this swarm is at r + (p/m) dt, p + F dt, t + dt, so we
should have
f(r + Jjj dt, p + F dt, t + dt)  f (r, p, t) = Q dt
the difference being the net gain of points in time dt. Expansion of the
first f in a Taylor's series, subtraction and division by dt results in
Eq. (132) again.
For a gas in equilibrium, under the influence of no external forces,
its molecules have a Maxwell distribution in momentum, f is independ
ent of t and of r. Since f must satisfy Eq. (132), we see that for this
case Q must be zero for all values of r and p. In other words, at
equilibrium, for every pair of molecules whose points are in a given
region of phase space and which collide, thus disappearing from the
region, there is another pair of molecules with momentum just after
collision such that they appear in the region; for every molecule which
loses a given momentum by collision, there is somewhere another mol
ecule which gains this momentum by collision.
Now suppose the gas is under the influence of a conservative force,
representable by a potential energy (p(r), such that F = grad r <p. If
the gas is in equilibrium, we would expect that the balance between
loss and gain of points still held and that Q is zero everywhere. How
ever now the density of the gas may differ in different parts of the con
tainer, so f may be a function of r as well as p. At equilibrium,
though, the dependence of f on p should still be Maxwellian, since the
temperature is still uniform throughout the gas. In other words the
distribution function should have the form f(r,p,t) = f r (r) • fp(p), where
factor fp will have the Maxwellian form of Eq. (127).
Therefore for a gas in equilibrium at temperature T in the pres
ence of a potential field <£>, the Boltzmann equation will be
(p/m) • grad r f  (grad r (p) • (grad p f ) =
Because f = f r f p and since f p is given by Eq. (127) we have grad p (f)
= (p/mkT)f r f p , so the equation further simplifies,
106 KINETIC THEORY
(p/m) • [grad r f r + (f r /kT) grad r 0]f p =
or
grad r f r = (f r /kT)grad r (p or f r (r) = Be ^ kT
Thus the distribution function for this case is
f(r ' p) = (2™kT)V 2 exp I " w (£ + *)
(133)
(134)
This is known as the MaxwellBoltzmann distribution. Constant B
must be adjusted so that the integral of f over all allowed regions of
phase space is unity. The formula states that the probability of pres
ence of a molecule at a point r,p in phase space is determined by its
total energy (p 2 /2m)+(p(r) = H(r,p); the larger H is, the smaller is
the chance that a molecule is present; the smaller T is, the more
pronounced is this probability difference between points where H dif
fers.
A Simple Example
Suppose a gas at equilibrium at temperature T is confined to two
interconnected vessels, one of volume V 1? at zero potential energy,
the other of volume V 2 at the lower potential energy (p= y . The
connecting tube between the two containers should allow free passage
of the gas molecules, although its volume should be negligible com
pared to V\ or V 2 . The potential difference between the vessels may
be gravitational, the vessels being at different elevations above sea
level; or it may be due to an electric potential difference, if the mol
ecules possess electric charges.
In this simple case the factor f r = Be~0/kT of the MaxwellBoltz
mann distribution is simply B throughout volume V\ and Be^A T
throughout V 2 . Since the integral of ff_ over the total volume must be
unity, we must have B = [V l + V 2 e?AT]i # The number of molecules
per unit volume in the upper container V^ is B times the total num
ber N of molecules in the system; the density of molecules in the
lower container V 2 is NBe?A T . Since the distribution in momentum
is Maxwellian in both vessels, the pressure in each container is kT
times the density of molecules there,
NkT
For V x : P x = ~ +  e y/kT
mV 2 :P 2 ^ Vie ,^ T + v2 (135)
THE MAXWELLBOLTZMANN DISTRIBUTION 107
At temperatures high enough so that kT 7$> y the exponentials in
the denominators of these expressions are nearly unity and the pres
sures in the two vessels are both roughly equal to kT times the mean
density N/(V 1 + V 2 ) of molecules in the system. If the temperature is
less than (y/k), however, the pressures in the two vessels differ ap
preciably, being greater in the one at lower potential, V 2 . When kT
<^C y practically all the gas is in this lower container.
The mean energy of a point atom in the upper container is all ki
netic, and thus is (3/2)kT; the mean energy of a point atom in the
lower vessel is (3/2)kT plus its potential energy there, y . Conse
quently, if the molecules are point atoms, having only translational
kinetic energy, the total energy of the system is the sum of the num
ber of molecules in each vessel times the respective mean energies,
U = NBVJ kTJ + NBV 2 e^ kT ( kT
3 NyV 2
=  NkT *£ 136)
2 V x e" kT + V 2
which changes from N[(3/2)kT  y) when kT <C y and all the gas is
/3 V2> \
in V 2 to N(kT  ' ) when kT ^> y and the density is practi
cally the same in both containers.
To compute the entropy and other thermodynamic potentials we
must first recognize that there are three independent variables, T,
Vj, and V 2 . The appropriate partials of S can be obtained by the pro
cedures of Chapter 8,
/ as \ = Cv = i. / au\ /_as \ = ( dPA
V3t; ViVi  t ~t[btJ v ^ w TV2 "Ut; v
etc.
iV,
Therefore,
3 /T\ 3 /v,+V,e>/ kT
2 Nk HtJ + l Nk + s o + Nk ln [^r —
(N ^ k 7 T T) (137)
v ie >/ kT + v 2
3_, /T\ _ , /v i + V 2 e>/ kT
IS TS  NkT lnl^l  NkT In.
©
108 KINETIC THEORY
where S and V are constants of integration.
The heat capacity of the gas at constant V x and V 2 ,
_ 3 (Nr»V 1 V 8 A'P)e>^ kT
C V lV2 = 2 Nk ( Vl e>AT + v 2 ) 2
is (3/2)kN both for kT <C > and for kT » > , but for intermediate
temperatures C v is larger than this. At low temperatures nearly all
the molecules are in the lower vessel and additional heat merely
speeds up the molecules; at temperatures near y/k the added heat
must push more molecules into the upper vessel as well as speed them
all up; at very high temperatures the density in the two containers is
nearly equal and additional heat again serves merely to increase ki
netic energy. We also note that P x = (dF/dVj), and similarly for P 2 ,
as required by Eqs. (88).
A More General Distribution Function
The form of the Maxwell Bo ltzmann distribution suggests some
generalizations. In Eq. (134) the expression in the exponent is the
total energy of position and of motion of the center of mass of the
molecule, and f itself is the probability density of position and mo
mentum of the center of mass. One obvious generalization is to put
the total energy of the molecule in the exponent and to expect that the
corresponding f is the probability density that each of the molecular
coordinates and momenta have specified values. The position coordi
nates q need not be rectangular ones, they may be angles and radii,
or other orthogonal curvilinear coordinates. We specify the nature of
these coordinates in terms of their scale factors h, such that hj dq^
represents actual displacement in the q^ direction (as r d6 is dis
placement in the 6 direction). For rectangular coordinates h is unity;
for curvilinear coordinates h may be a function of the q's.
Suppose each molecule has v degrees of freedom; then it will need
v coordinates q 15 q 2 , ... , q^ to specify its configuration and position
in space. If the coordinates are mutually perpendicular and if hj is
the scale factor for coordinate qj, the volume element for the q's is
dVq = h x dq x h 2 dq 2 ••• h v dqj, and the kinetic energy of the molecule is
, v
<K.E.>= £ m^fq?, q. = dq^dt (138)
i = 1
where m^ is the effective mass for the ith coordinate (total mass or
reduced mass or moment of inertia, as the case may be).
Following the procedures of classical mechanics we define the mo
mentum P, conjugate to q if as
THE MAXWELLBOLTZMANN DISTRIBUTION 109
Pi = ^ (K.E.) = m.hie.j
We now define the Hamiltonian function for the molecule as the total
energy of the molecule, expressed in terms of the p's and q's,
H(p,q)= £ o^:(PiAi) 2 + 0(q) (139)
i = 1 Zmi
where (p is the potential energy of the molecule, expressed in terms
of the q's. The h's may also be functions of the q's, but the only de
pendence of H on the p's is via the squares of each p^, as written
specifically in the sum. It can then be shown that the corresponding
scale factors for the momentum coordinates, the other half of the
2v dimensional phase space for the molecule, are the reciprocals of
the h's, so that the momentum volume element is (dp 1 /h 1 )(dp 2 /h 2 ) •••
Idpp/hj,) = dV p .
As an example, consider a diatomic molecule, with one atom of
mass m x at position x 1 ,y 1 ,z 1 and another of mass m 2 at X2,y 2 ,z 2 . We
can use, instead of these coordinates, the three coordinates of the cen
ter of mass, x = [(m^ + m 2 x 2 )/(m 1 + m 2 )] and similarly for y and z,
plus the distance r between the two atoms and the spherical angles 9
and cp giving the direction of r. Then the total kinetic energy of the
molecule, expressed in terms of the velocities, is
± (m,+ m 2 )(x 2 + y 2 + z 2 ) +   2 (r 2 + r 2 9 2 + r 2 sin 2 9 <p 2 )
z m » < m 2
so that the volume element dVq = dx dy dz dr r d6 r sin 6 d<p. The
momenta are
Px = ( m i + m 2^ etc > P r = m r r p^ = m r r 2 ^'
p^= m r r 2 sin 2 9<p
where m r is the reduced mass [m 1 m 2 /(m 1 + m 2 )]. The kinetic energy
expressed in terms of the p's is
2(m/ + m 2 ) t p x + p y + p z ] + 2^
Pr
r L
and the volume element dV p = dp x dp y dp z dp r (dp^/rJCdp^/r sin 9).
For a molecule with v degrees of freedom, the distribution function
is
110 KINETIC THEORY
f(q,P)  ry ry e
^q ^p
"\f )*!// "
where
Zq = / ... /e ^)AT dVq
Z P = J '" / ex p
f. p!A!"
>' 2m; kT
L i = 1 i J
(1310)
dV D = (27TkT) (1/2)j 'Vm 1 m 2 m i ,
and where f(q,p) dV q dV p is the probability that the first molecular
coordinate is between q 1 and q x + dq 1? the second is between q 2 and
q 2 + dq 2 , and so on, that the first momentum coordinate lies between
p x and p x + dp 1? and so on. Since the scale factors h^ are not func
tions of the p's, they enter as simple constants in the integration over
the p's and thus the normalizing constant Zp can be written out ex
plicitly. We can also compute explicitly the mean total kinetic energy
of the molecule, no matter what its position or orientation:
<K.E.>
Pi
total = ^ / J e ^ AT dv q 5p S "' /?^lq
v pi a!
! 4 2mTkT j dV P
fa^);/"v/& «*(!
x dui du 2 ••• dui, = kT (1311)
Mean Energy per Degree of Freedom
Therefore each degree of freedom of the molecule has a mean ki
netic energy (l/2)kT, no matter whether the corresponding coordinate
is an angle or a distance and no matter what the magnitude of the mass
m^ happens to be. The thermal energy of motion of the molecule is
equally distributed among its degrees of freedom. The mean value of
the potential energy of course depends on the nature of the potential
function (p(q), although a comparison with the kineticenergy terms
indicates that if the sole dependence of (p on coordinate qj is through
a quadratic term (l/2)m i co?q then the mean potential energy for this
coordinate is also (l/2)kT (see below).
This brings us to an anomaly, the resolution of which will have to
await our discussion of statistical mechanics. A diatomic molecule
THE MAXWELLBOLTZMANN DISTRIBUTION 111
has six degrees of freedom (three for the center of mass, one for the
interatomic distance, and two angles for the direction of r, as given
above) even if we do not count the electrons as separate particles.
We should therefore expect the total kinetic energy of a diatomic gas
to be the number of molecules N times six times (l/2)kT and thus U,
the internal energy of the gas, to be at least 3NkT (it should be more,
for there must be a potential energy, dependent on r, to hold the mol
ecule together, and this should add something to U). However, meas
urements of heat capacity C v = (8U/8T) V show that just above its boil
ing point (about 30° K) the U of H 2 is more nearly (3/2)NkT, that be
tween about 100° K and about 500° K the U of H 2 , O z , and many other
diatomic gases is roughly (5/2)NkT and that only well above 1000° K
is the U for most diatomic molecules equal to the expected value of
(6/2)NkT (unless the molecules have dissociated by then). The rea
sons for this discrepancy can only be explained in terms of quantum
theory, as will be shown in Chapter 22.
A Simple Crystal Model
Hitherto we have tacitly assumed that the coordinates q of the
molecules, as used in Eq. (139), are universal coordinates, referred
to the same origin for all molecules. This need not be so; we can re
fer the coordinates of each molecule to its own origin, provided that
the form of the Hamiltonian for each molecule is the same when ex
pressed in terms of these coordinates. For example, each atom in a
crystal lattice is bound elastically to its own equilibrium position,
each oscillating about its own origin. The motion of each atom will af
fect the motion of its neighbors, but if we neglect this coupling, we can
consider each atom in the lattice to be a threedimensional harmonic
oscillator, each with the same three frequencies of oscillation (in
Chapter 20 we shall consider the effects of the coupling, which we
here neglect). In a cubic lattice all three directions will be equivalent,
so that (without coupling) the Hamiltonian for the jth atom is
h j = 25, Hj + p 2 yj + e 2 zj > + \ ™* (*} + y) + V (13 ' 12)
where xj is the x displacement of the jth atom from its own equili
brium position and p x j is the corresponding mxj. Thus expressed, H
has the same form for any atom in the lattice.
In this case we can redefine the MaxwellBoltzmann probability
density as follows: (l/ZqZ p ) exp (Hj/kT) is the probability density
that the jth atom is displaced xj ,yj ,zj away from its own equilibrium
position and has momentum components equal to Pxj>Pyj>Pzj Tne nor
malizing constants are [using Eqs. (126)]
112 KINETIC THEORY
3/2
Z q
mu>V/2kT , I 3 _r2irkT
j e ma,x/ Z KT dx
_ ~
mw
Z D = (27rmkT) 3/2
J P
Therefore the probability density that any atom, chosen at random, has
momentum p and is displaced a distance r from its position of equi
librium in the lattice is
= / "
>3
f(r ' p) = to ex ?
2 2 2
p murr
2mkT 2kT
(1313)
for the simplified model of a crystal lattice we have been assuming.
We can now use this result to compute the total energy U, kinetic
and potential, of the crystal of N atoms, occupying a volume V, at
temperature T. In the first place we note that even at T = the crys
tal has potential energy of overall compression, when it is squeezed
into a volume less than its equilibrium volume V . If the compressi
bility at absolute zero is k, this additional potential energy is
(V  V ) 2 /2kV ; the potential energy increases whether the crystal is
compressed or stretched. We should also notice that the natural fre
quencies of oscillation of the atoms also are affected by compression;
u) is a function of V.
The thermal energy of vibration of a typical atom in this crystal is
given by the integral of Hf over all values of r and p. Again using
Eqs. (126) we have
/... /Hf dV p dV q = ( ^!L_ / p2e P 2 / 2mkT dp
J J P q mV277mkT 
, 3 2 /mo; 2 [° 2 mco 2 x 2 /2kT .
+ 2™ywi xe *
= kT + kT = 3kT = 3(R/N )T (1314)
showing that the mean thermal kinetic energy per point particle is
(3/2)kT whether the particle is bound or free and that the mean ther
mal potential energy of a threedimensional harmonic oscillator is
also (3/2)kT, independent of the value of co.
Thus, for this simple model of a crystal, with N = nN ,
U = 3nRT +[(VV ) 2 /2kV ]; C v = 3nR =T(aS/3T) v (1315)
One trouble with this simple model is that it does not provide us
with enough to enable us to compute the entropy. We can obtain
(3S/aT) v from Eq. (1315) but we do not know the value of (3S/3V) T =
THE MAXWELLBOLTZMANN DISTRIBUTION 113
(3P/3T) V [see Eq. (812)], unless we assume the equation of state
(91) and set (3P/3T) V =0/k from it. But an even more basic defi
ciency is that it predicts that C v = 3nR for all values of T. As shown
in Fig. 31 and discussed prior to Eq. (92), C v is equal to 3nR for
actual crystals only at high temperatures; as T goes to zero C v goes
to zero. We shall show in Chapter 20 that this behavior is a quantum
mechanical effect, related to the anomaly in the heat capacities of di
atomic molecules, mentioned in the paragraph after Eq. (1311).
Magnetization and Curie's Law
Another example of the use of the generalized Maxwell Bo ltzmann
distribution is in connection with the magnetic equation of state of a
paramagnetic substance. Such a material has permanent magnetic di
poles of moment [i connected to its constituent molecules or atoms.
If a uniform magnetic field ($> = jj. 3C ( jU is not a moment, it is the
permeability of vacuum) is applied to the material, each molecule will
have an additional potential energy term ju(B cos 6 in its Hamiltonian.
where 6 is the angle between the dipole and the direction of the field
(and jll is the dipole 's moment). We note that the (B and 3C used here
are those acting on the individual molecules, which differ from the
fields outside the substance.
The distribution function, giving the probability density that the
molecule has a given momentum, position, and orientation, can be
written
, e ■' o , ,., ,„ x u(B cos 0/kT
f = f P fqfe; ffl =(l/Z )e^ '
where fp is the momentum distribution, so normalized that the inte
gral of fp over all momenta is unity; fq is the position distribution
for all position coordinates except 6, the orientation angle of the
magnetic dipole (fq is also normalized to unity). Thus the factor
Iq gives the probability distribution for orientation of the magnetic
dipole; each dipole is most likely to be oriented along the field (6 = 0)
and least likely to be pointed against the field (6 = 77). When kT ^> jllcb
the difference is not great; the thermal motion is so pronounced that
the dipoles are oriented almost at random. When kT ^C,u.(B the differ
ence is quite pronounced. Nearly all the dipoles are lined up parallel
to the magnetic field; the thermal motion is not large enough to knock
many of them askew.
The factor Iq can be normalized separately,
r jii(B cos 6/kT „ An /2kT\ . . *x<B
Z/q = / e ^ ' sin 6 d6 = [ sinh j=
y \u(B/ kT
114
KINETIC THEORY
and can be used separately to find the mean value of the component
jul cos of the dipole moment along the magnetic field. This, times the
number of dipoles per unit volume, is, of course, the magnetic polari
zation <? of the material, as defined in Chapter 3. And, since the mag
netization is an = jLi V(P, we have
am
^/(Hcos0)e
/i(B cos 6/kT
sin 6 d6
(1316)
•0 °
= Nm/Xo
= NjLLjLLo
(2/x)cosh x  (2/x 2 ) sinh x
jU(B
(2/x) sinh x ' ^T
f(NjLL {A 2 (B/3kT), kT>M(B
InjUoM, kT«CM(B
:©)(sj
Thus we see that at low fields (or high temperatures) the dipoles tend
only slightly to line up with the field and the magnetization 9H is pro
portional to (B = jLL 3C, but that at high magnetic intensities (or low
temperatures) all the dipoles line up and the magnetization reaches
its asymptotic value NMoM (i.e., it is saturated), as shown in Fig.
131. Since the magnetic moment of an oxygen molecule is roughly
kT//xB
ffl II I I I [
i i i i
J I I I I MM
I I I I I Mil
MINI
o.i
10
fiB/kt
FIG. 131. Magnetization curve for paramagnetic substances.
THE MAXWELLBOLTZMANN DISTRIBUTION 115
3xl0" 23 mks units, /KB is no larger than 3xl0 23 joules for <B = 1
weber per m 2 (= 10,000 gauss, a quite intense field). Since kT
= 3x 10" 23 joules for T  2° K, the parameter x = jxce/kT is consid
erably less than unity for 2 (for example) for temperatures greater
than 30° K and/or (B less than 1 weber/m 2 . In such cases, where
x<l, the polarization (P is much smaller than 3C, so that the 3C act
ing on the molecule is not much different from the X outside the ma
terial.
Thus for most temperatures and field strengths, for paramagnetic
materials like 2 , x is very small and Curie's law
9TC=nD3C/T; D = N MoM 2 /3k (1317)
is a good approximation for the magnetic equation of state. Kinetic
theory has thus not only derived Curie's law [see Eq. (38)] and ob
tained a relation between the Curie constant D and the molecular
characteristics of the material (such as p. and the basic constants N
jllq, and k) but has also determined the limits beyond which Curie's
law is no longer valid, and the equation of state which then holds.
For example, for the paramagnetic perfect gas of Chapter 7, the
more accurate equation for T dS and the adiabatic formula (714) is
TdS=[
nR  ^ (x 2 csch 2 x  1) dT  iSi. dP
3nD . 2 _ U 2 _ <\\ .™ nRT
3nDT
a 2
and
T 5/2
' "c^lc (x ' csch2 x " X) d3C
n 3D/Ra 2
T . . [aw\ ' 3D3C ^ u (ax\
const.
(1318)
where x = a3C/T and a  jLtpio/k. This reduces to Eq. (714) when x
is small.
This completes our discussion of paramagnetic materials. In the
case of ferromagnetic materials, where the polarization (P is not
small compared to H, the field acting on the individual dipole is con
siderably modified by the polarization of the nearby dipole; in fact the
dipoles may tend to line up all by themselves. But it would go too far
afield to discuss permanent magnetization.
Transport
Phenomena
The Boltzmann equation (132) can also be used to calculate the
progress of some spontaneous processes, such as the mixing of two
gases by interdiffusion and the attenuation of turbulence in a gas by
the action of viscosity. All these processes have to do with the trans
port of something, foreign molecules or electric charge or momentum,
from one part of the gas to another; consequently they are called
transport phenomena. Here, because the system is not in equilibrium,
the collision term Q, which measures the net rate of entrance and
exit of molecular points in phase space, is not zero. An exact calcu
lation of the dependence of Q on p and r is quite a difficult task, re
quiring a detailed knowledge of molecular behavior during a collision.
Exact expressions for Q have been determined for only a few, simple
cases. Luckily there is a relatively simple, approximate expression
for Q, which will be good enough for the calculations of this chapter.
An Approximate Collision Function
The function Q in the Boltzmann equation represents the effects of
collisions in bringing the gas into equilibrium. As we pointed out
earlier, when the gas is in equilibrium Q is zero; as many molecules
gain a given momentum per second by a collision as there are those
that lose this momentum per second by a collision. And if the gas is
close to equilibrium Q should be small. Suppose the solution of the
Boltzmann equation for equilibrium conditions is fo(r,p) and the solu
tion for nonequilibrium conditions close to the equilibrium state is
f(r,p). What we have just been saying, in regard to the behavior of Q
near equilibrium, is that it should be proportional to f  f , when f is
nearly equal to f . A glance at Eq. (132) indicates that the propor
tionality constant has the dimensions of the reciprocal of time and can
be written as l/t c (r,p), where t c is called the relaxation time of the
system, for reasons shortly to be apparent. We will thus assume that
the Boltzmann equation, for conditions near equilibrium, can be written
116
TRANSPORT PHENOMENA 117
(3f/9t) + (p/m) • grad r f + F • grad p f = [(f  f)/t c ]
where fo is the distribution for the nearest equilibrium state, and where
f o  f is small compared to either f or f . Therefore the lefthand side of
this equation is also small, and it would not produce additional firstorder
error to substitute f for f on the lefthand side. Thus an approximate
equation for nearequilibrium conditions is
f fo  t c [(3fo/3t) + (p/m) • grad r f + F ■ grad p f ] (141)
Collisions between gas molecules are fairly drastic interruptions of the
molecule's motion. After the collision the molecule's direction of motion,
and also its speed, may be drastically altered. Roughly speaking, it is as
though each collision caused the participating molecule to forget its pre
vious behavior and to start away as part of an equilibrium distribution of
momentum; only later in its free time, before its next collision, does the
nonequilibrium situation have a chance to reaffect its motion.
For example, we may find a gas of uniform density having initially a
distribution in momentum f^(p ) of its molecules which differs from the
Maxwell distribution (127) ; there may be more fast particles in relation
to slow ones than (127) requires, or there may be more going in the x
direction than in the y, or some other asymmetry of momentum which is
still uniform in density. Such a distribution f^ since it is not Maxwellian,
is not an equilibrium distribution. However, since the lack of equilibrium
is entirely in the "momentum coordinate" part of phase space, the distri
bution can return to equilibrium in one collision time; we would expect that
at the next collision each molecule"would regain its place in a Maxwell dis
tribution, so to speak. The molecules do not all collide at once, the chance
that a given molecule has not had its next collision, after a time t, is
e~V T , where t is the mean free time between collisions [see Eqs. (1111)
and (1211)]. Thus we would expect that our originally anisotropic distri
bution would "relax" from f i back to f with an exponential dependence
on time of e _t / T (note that t is proportional to X, the molecular mean
free path).
But if f is independent of r and there is no force F acting, a solution
of Eq. (141) which starts as f = f A at t = is
f = f +(fi  fJeVtc (142)
which has just the form we persuaded ourselves it should have, except
that the relaxation time t is in the exponent, rather than the mean free
time t= < mX/p> . We would thus expect that the relaxation time t c , en
tering Eq. (141) would be approximately equal to mA/p. Detailed calcula
tions for the few cases which can be carried out, plus indirect experimen
tal checks (described later in this chapter) indicate that it is not a bad ap
proximation to set t c = < mA/p> = r. This will be done in the rest of this
chapter.
118 KINETIC THEORY
Electric Conductivity in a Gas
Suppose that a certain number N^ of the molecules of a gas are
ionized (N^ being small compared to the total number N of molecules)
and suppose that initially the gas is at equilibrium at temperature T.
At t = a uniform electric field 8, in the positive x direction, is
turned on. The ions will then experience a force es in the x direc
tion, where e is the ionic charge. Imposed on the random motion of
the ions between collisions will be a "drift" in the x direction. This
is not an equilibrium situation, since the drift velocity of the ions will
heat up the gas. But if Ni/N is small, and if 8 is small enough, the
heating will be slow and we can neglect the term 3f/3t in Eq. (141)
in comparison with the other terms.
Since the ions are initially uniformly distributed in space and
since the ionic drift is slow, we can assume that f is moreorless
independent of r. Thus Eq. (141) for the ions becomes
f  fo  t c F • grad p f = f  et c 8 (9f /8p x ) « [ 1 + et c 8 /mkT)p x J f
(143)
where f is the Maxwell distribution,
3/2
f  [l/(2 7rmkTr 2 ] exp
(PxPyPz)
2mkT
of the neutral molecules. Function f will be a good approximation to
the correct momentum distribution of the ions if the second term in
the brackets of Eq. (143) is small compared to the first term, unity,
over the range of values of p x for which f has any appreciable mag
nitude. The term et c Sp x /mkT can be written (eA8/kT)(p x < l/p>)
if we assume that t c = r = < mA/p>, A being the mean free path of
the molecule [see Eq. (1211)]. Since eA8 is the energy that would
be gained by the ion (in electron volts, if the ion is singly ionized) by
falling through a mean free path in the direction of 8, and since kT
in electron volts is T/7500, then for a gas (such as 2 ) at standard
conditions, where A^ 10~ 7 m [see the discussion following Eq. (1210)]
and T300, the factor eAE/kT^ 8/40,000, 8 being in volts per me
ter. Thus if 8 is as large as 4000 volts per meter, the second term
will not equal the first in Eq. (143) until p x is 10 times the mean
momentum <p> and by this time the exponential factor of f will
equal about e~ 50 . Thus, for a wide range of values of T and of 8,
either f is vanishingly small or else the second term in brackets of
Eq. (143) is small compared to the first.
What Eq. (143) indicates is that the momentum distribution of the
ions, in the presence of the electric field, is slightly nonisotropic;
somewhat more of them are going in the direction of the field (p x pos
TRANSPORT PHENOMENA 119
itive) than are going in the opposite direction (p x negative). There is
a net drift velocity of the ions in the x direction:
r 7 /■ rrr/Px 2et c 8 Px \
U X = U I (p x /m)f dV p  U f (£ + ^ ) f (p ) dV p
«. !*£! M £H = Aeg Mf , _^i_ , MS (144)
m m WmkT/ m<v>
where we have used the fact that for a Maxwell distribution f , <p x >
= and <p x /2m> = (l/2)kT and we have also used Eq. (1211) for
the mean free time t.
We see that the drift velocity U of the ion is proportional to the
electric intensity, as though the ion were moving through a viscous
fluid. The proportionality factor M — et c /m ^Xe/<v> is called the
mobility of the ion. The current density I = (N^ell/V) (in amperes per
square meter) is
I « (NiAe 2 /V)(2/77mkT) l/2 8 (145)
obeying Ohm's law, with a conductivity NjeM/V = Nje 2 t c /mV.
Drift Velocity
It is interesting to see that the drift velocity, and therefore the cur
rent density, is proportional to the mean free time between collisions
and is thus inversely proportional to the square root of the tempera
ture T. As T increases, the random velocity <v> of the ions (and
neutral molecules) increases, but the drift velocity U of the ions de
creases. One can visualize the process by imagining the flight of an
ion from one collision to its next. Just after each collision the ion
comes away in a random direction, with no initial preference for the
direction of s. But during its free flight the electric field acts on it,
turning its motion more and more in the positive x direction (its path
is a portion of a parabola) and thus adding more and more positive x
component to its velocity. This accentuation of the positive x motion
is completely destroyed by the next collision (on the average) and the
molecule starts on a new parabolic path. If the mean free time is long,
the molecule has plenty of time to add quite a bit of excess v x ; if t
is small, the molecule hardly has time to get acted on by the field be
fore it collides again. Thus the higher the temperature, the greater
the random velocity < v> , the shorter the mean free time t and the
smaller the drift velocity and current density. This, of course, checks
with the measurements of gaseous conduction.
In most ionized gases, free electrons will be present as well as
positive ions. The electrons will also have a drift velocity, mobility,
120 KINETIC THEORY
and current density, given by Eqs. (144) and (145), only with a
negative value of charge e, a different value of A and a much
smaller value of mass m. Thus the drift velocity will be oppo
site in direction to that of the ions but, since the charge e is neg
ative, the current density is in the same direction as that of the
ions. Since the electronic mean free path is roughly 2 to 4 times
that of the ions and since the electronic mass is several thousand
times smaller, the electronic mobility is 500 to 1000 times greater
than that of the positive ions and therefore most of the current in an
ionized gas is carried by the electrons.
Diffusion
Another nonequilibrium situation is one in which different kinds of
molecules mix by diffusion. To make the problem simple, suppose we
have a small number Ni of radioactive "tagged" molecules in a gas
of N nonradioactive molecules of the same kind. Suppose, at t = 0,
the distribution in space of the tagged molecules is not uniform (al
though the density of the mixture is uniform). Thus the distribution
function for the tagged molecules is a function of r, and we have to
write our "0th approximation" as
— p 2 /2mkT
f o= f r(r)ip(p); hJ^^fT'> //.ArdVr = l
(146)
The distribution function f for the diffusing molecules will change
with time, but we will find that the rate of diffusion is slow enough so
that the term 9f/3t in Eq. (141) is negligible compared to other
terms.
In the case of diffusion there is no force F, but f does depend on
r, so the approximate solution of Eq. (141) is
f  f  t c (p/m) • grad r f = [f r  (t c /m)p • grad r f r ]f p (147)
where the vector grad r f r points in the direction of increasing density
of the tagged molecules. The anisotropy is again in the momentum
distribution, but here the preponderance is opposite to grad r f r , there
is a tendency of the tagged molecules to flow away from, the region of
highest density. The conditional probability density that a molecule, if
it is at point r, has a momentum p, is [see Eq. (112)]
(f/f r ) [1  (t c /m)pg]f p ; g = (l/f r )grad r f r
From this we can compute the mean drift velocity of the tagged mole
cules which are at point r (for convenience we point the x axis in the
direction of g):
TRANSPORT PHENOMENA 121
oo
U  Iff (p/m)[l  (t c /m)p x g]f p dV p = 2(t c /m)g
oo
x J j /'( P y2m)f p dV  (t c kT/m)g  A(2kT/7rm) V2 g
— OO "
* 2A<v>(l/f r )grad r f r
We see that in this case the drift velocity increases as T increases.
The density p^ of tagged molecules at r is Nif r molecules per
unit volume, so the flux J of tagged particles at r, the net diffusive
flow caused by the uneven distribution of these particles, is
J = Nif r U ^Dgrad r Pi; D = t c kT/m « A (2kT/nm) l/2 (148)
where constant D is called the diffusion constant of the tagged mole
cules. A density gradient of tagged molecules produces a net flow
away from the regions of high density, the magnitude of the flow being
proportional to the diffusion constant D. We note that there is a sim
ple relationship between D and the mobility M of the same molecule
when ionized and in an electric field, as given in Eq. (144),
D = (kT/e)M (149)
which is more accurate than our approximation for t c .
Thus a measurement of diffusion in a gas enables us to predict the
electrical conductivity of the gas or vice versa. Equation (148) is the
basic equation governing diffusion. By adding to it the equation of con
tinuity, we obtain
dpi/dt = div J * DV 2 pi (1410)
which is called the diffusion equation.
There are a number of other transport problems, heat flow and
viscosity, for example, which can be worked out by use of the Boltz
mann equation. These will be given as problems.
Fluctuations
Any system in thermal equilibrium with its surroundings undergoes
fluctuations in position and velocity because of the thermal motion of
its own molecules as well as of any molecules that may surround it.
Kinetic theory, which enables us to compute the mean thermal kinetic
and potential energy inhering in each degree of freedom of the system,
makes it possible to compute the variance (i.e., the meansquare am
plitude of the fluctuations) of each coordinate and momentum of the
system. It is often useful to know the size of these variances, for they
tell us the lower bound to the accuracy of a piece of measuring equip
ment and they sometimes give us a chance to measure, indirectly, the
magnitude of some atomic constants, such as Avogadro's number N .
Equipartition of Energy
Referring to Eqs. (139) and (1310), we see that if the Hamiltonian
function for a system can be separated into a sum of terms [(l/2m^)
x (pi/hj) 2 + 9i(qj)J, each of which is a function of just one pair of vari
ables, p^ and q^ , then the Maxwell Boltzmann probability density can
be separated into a product of factors, (1/Z^) exp[(l/2mikT)(pi/hi) 2
 (l/kT)(p^(qi)\ , each of which gives the distribution in momentum
and position of one separate degree of freedom. Even if the potential
energy, or the scale factors h, cannot be completely separated for
all the degrees of freedom of the system, if the potential energy does
not depend on some coordinates qj (such as the x coordinates of the
center of mass of a dust particle floating freely in the air), then all
values of that coordinate are equally likely (the dust particle can be
anywhere in the gas) and its momentum will be distributed according
to the probability density
fpj(Pj> = (1/Zj)exp [ 2^jkT (f IT ; Z J = < 2,rm J kT > l/2 < 15 V
122
FLUCTUATIONS 123
The mean thermal kinetic energy of the jth degree of freedom is
thus
OO
<K.E.> = /(p/2m j h])f pj (dpj/h j )=ikT (152)
whether the coordinate is an angle or a distance or some other kind of
curvilinear coordinate. Therefore the kinetic energy of thermal mo
tion is equally apportioned, on the average, over all separable degrees
of freedom of the system, an energy (l/2)kT going to each. If the po
tential energy is independent of qj, then (l/2)kT is the total mean en
ergy possessed by the jth degree of freedom. On the average the en
ergy of rotation of a diatomic molecule (described in terms of two
angles) would be kT and the average energy of translation of its cen
ter of mass would be (3/2)kT. A light atom (helium for example) will
have a higher mean speed than does a heavy atom (xenon for example)
at the same temperature, in order that the mean kinetic energy of the
two be equal. In fact the mean square value of the jth velocity, when
qj is a rectangular coordinate (i.e., when hj = 1), is
<qj > = <Pj/ m j> = k T/mj (153)
Mean Square Velocity
For example, the x component of the velocity of a dust particle in
the air fluctuates irregularly, as the air molecules knock it about. The
average value of x is zero (if the air has no gross motion) but the
mean square value of x is just kT divided by the mass of the parti
cle. We note that this meansquare value is independent of the pres
sure or density of the air the particle is floating in, and is thus inde
pendent of the number of molecules which hit it per second. If the gas
is rarefied only a few molecules hit it per second and the value of x
changes only a few times a second; if the gas is dense the collisions
occur more often and the velocity changes more frequently per second
(as shown in Fig. 151), but the mean square value of the velocity is
the same in both cases if the temperature is the same.
Even if the potential energy does depend on the coordinate qj, the
mean kinetic energy of the jth degree of freedom is still (l/2)kT. If
the potential energy can be separated into a term 0j(qj) and another
term which is independent of qj, then the probability density that the
jth coordinate has a value qj is
fqjtaj) = (l/Z qj )e"^ /kT ; Z qj = /e~^ /kT hj d qj (154)
124
KINETIC THEORY
Fig. 151. Variation with time of x component of velocity
and displacement of Brownian motion. Lower
curves for mean time between collisions five
times that for upper curves.
where the integration is over the allowed range of qj. The mean value
of the potential energy turns out to be a function of kT, but the nature
of the function depends on how </>j varies with qj.
The usual case is the one where the scale constant is unity and
where the potential energy <pi = (l/2)m. a^q^ has a quadratic depend
•* J J J
ence on qj, so that in the absence of thermal motion the displacement
q^ executes simple harmonic motion with frequency uji/2it. In this
case Z a i = (27ikT/m i co^ ) l/2 , and the mean value of potential energy
J qj
J W J
when thermal fluctuations are present is
<^>
1 2
2 m i w
1
/q]f qj dq rt kT
(155)
Thus when the potential energy per degree of freedom is a quadratic
function of each q, the mean potential energy per q is (l/2)kT, inde
pendent of cdj, and equal to the mean kinetic energy, so that the total
mean energy for this degree of freedom is kT. Also the meansquare
displacement of the coordinate from its equilibrium position is
<q]>
kT/m jW j
FLUCTUATIONS 125
for coordinates with quadratic potentials.
As an example of a case where <p, is not quadratic, we recall the
case of the magnetic dipoles, where 0j = /i® cos 6 , so that Z q j
= (2kT//Lt(B) sinh (/i(B/kT). From Eq. (1316) the mean potential energy
is
<  ii (B cos 6 > = kT  )Li(B coth (m (B/kT) ~~
( M 2 (B 2 /kT), kT»^(B
jll(B + kT, kT«C /i(B
which is not equal to (l/2)kT.
Fluctuations of Simple Systems
A mass M, on the end of a spring of stiffness constant K, is con
stantly undergoing small, forced oscillations because of thermal fluc
tuations of the pressure of the gas surrounding it and also because of
thermal fluctuations of the spring itself. In the absence of these fluc
tuations the mass will describe simple harmonic motion of amplitude
A and frequency (1/2tt)(K/M) i/2 , so its displacement, and its mean
kinetic and potential energy would be
x = A cos[t(K/M) l/2 + a]; <K.E.> = ^KA 2 = <P.E.>
where A and a are determined by the way the mass is started into
motion. In the presence of the thermal fluctuations an irregular mo
tion is superposed on this steady state oscillation. Even if there is
no steady state oscillation the mass will never be completely at rest
but will exhibit a residual motion having total energy, potential plus
kinetic, of kT, having a meansquare amplitude such that (1/2)KA 2 T
= kT, or
A 2 T = 2kT/K .
With a mass of a few grams and a natural frequency of a few cycles
per second (K> 1), this mean square amplitude is very small, of the or
der of 10~ 2O m 2 , a root mean square amplitude of about 10" 8 cm. This is
usually negligible, but in some cases it is of practical importance. The
human eardrum, plus the bony structure coupling it to the inner ear,
acts like a mass spring system. Even when there is no noise present
the system fluctuates with thermal motion having a mean amplitude of
about 10" 8 cm. Sounds so faint that they drive the eardrum with less
amplitude than this are "drowned out" by the thermal noise. In actual
fact this thermalnoise motion of the eardrum sets the lower limit of
audibility of sounds in the frequency range of greatest sensitivity of
126
KINETIC THEORY
the ear (1000 to 3000 cps); if the incoming noise level is less than this
"threshold of audibility/' we "hear" the thermal fluctuations of our
eardrums rather than the outside noises.
We notice that the root mean square amplitude of thermal motion
of a mass on a spring, (2kT/K) l/2 , is independent of the density of the
ambient air and thus independent of the number of molecular blows
impinging on the mass per second. If the density is high the motion
will be quite irregular because of the large number of blows per sec
ond; if the density is low the motion will be "smoother," but the mean
square amplitude will be the same if the temperature is the same, as
illustrated in Fig. 152. Even if the massspring system is in a vac
uum the motion will still be present, caused by the thermal fluctua
tions of the atoms in the spring.
Time t ^
Fig. 152. Brownian motion of a simple oscillator for two
different mean times between collisions.
The same effect is present in morecomplex systems, each degree
of freedom has mean kinetic energy (l/2)kT and similarly for the po
tential energy if it depends quadratically on the displacement, as in a
simple oscillator. A string of mass p per unit length and length L
under tension Q can oscillate in any one of its standing waves; the
displacement from equilibrium and the total energy of vibration of the
nth wave is
FLUCTUATIONS 127
y n = A n sin(7inx/L) cos [(7rnt/L)(Q/p) l/2 + o n ]
i y
<K.E.> + <P.E.> = ^QA^vrn/L) 2 J sin 2 (7mx/L) dx
= i(7T 2 n 2 Q/L)A 2 n
When the string is at rest except for its thermal vibrations, each
of the standing waves has a mean energy of kT, so the meansquare
amplitude of motion A n of the nth wave is (4LkT/7r 2 n 2 Q) and the mean
square amplitude of deflection of some central portion of the string is
the sum of all the standing waves (because of the incoherence of the
motion, we sum the squares),
1 3N 3N
<y 2 > * \ Y] < A n> = (LkT/Q) VJ (l/7r 2 n 2 )  LkT/6Q
n= 1 n= 1
which is related to the result for the simple oscillator, 12Q/L being
equivalent to the stiffness constant K of the simple spring. If the
string is part of a string galvanometer these thermal fluctuations will
mask the detection of any current that deflects the string by an amount
less than (LkT/6Q) l/2 .
Density Fluctuations in a Gas
The thermal motion of the constituent molecules produces fluctua
tions of density, and thus of pressure, in a gas. We could analyze the
fluctuation in terms of pressure waves in the gas, as we analyzed the
motion of a string under tension in terms of standing waves. Instead
of this, however, we shall work out the problem in terms of the poten
tial energy of compression of the gas. Suppose we consider that part
of the gas which, in equilibrium, would occupy a volume V s and would
contain N s molecules. At temperature T the gas, at equilibrium,
would have a pressure P = N s kT/V s throughout. If the portion of the
gas originally occupying volume V s were compressed into a somewhat
smaller volume V s AV(AV <^CV S ), an additional pressure AP
= [N s kT/(V s  AV)]  P e* P(AV/V S ) would be needed and an amount of
work
AV
/ AP d(AV) = (P/V S )(AV) 2 = N s kT(AV/V s ) 2
o
would be required to produce this compression. When thus compressed,
this portion of the gas would have a density greater than the equilibrium
128 KINETIC THEORY
density p by an amount Ap = p (AV/V S ) and its pressure would be
greater than P by an amount AP  P(AV/V S ).
Therefore the potential energy corresponding to an increase of
density of the part of the gas originally in volume V s , from its equi
librium density p to a nonequilibrium density p + Ap, is (l/2)NkT
x (Ap/p) 2 . For thermal fluctuations the mean potential energy is
(l/2)kT, if the potential energy is a quadratic function of the variable
Ap(as it is here). Consequently the meansquare fractional fluctua
tion of density of a portion of the gas containing N s molecules (and
occupying volume V s at equilibrium) is
<(Ap/p) 2 > = l/N s (156)
which is also equal to the mean square fractional fluctuation of pres
sure, <(AP/P) 2 >. Another derivation of this formula is given in
Chapter 23.
We see that the smaller the fraction of the gas we look at (the
smaller N s is) the greater the fractional fluctuation of density and
pressure is caused by thermal motion. If we watch a small group of
molecules, their thermal motion will produce relatively large changes
in their density. On the other hand if we include a large number of
molecules in our sample, the large fluctuations in each small part of
the sample will to a great extent cancel out, leaving a mean square
fractional fluctuation of the whole which is smaller the larger the
number N s of molecules in the sample. The root mean square frac
tional fluctuation of density or pressure of a portion of the gas is in
versely proportional to the square root of the number of molecules
sampled.
These fluctuations of density tend to scatter acoustical and electro
magnetic waves as they travel through the gas. Indeed it is the scat
tering of light by the thermal fluctuations of the atmosphere which pro
duces the blue of the sky. The fluctuations are independent of temper
ature, although at lower temperatures the N s molecules occupy a
smaller volume and thus the fluctuations are more "finegrained" at
lower temperatures.
Incidentally, we could attack the problem of density fluctuations by
asking how many molecules happen to be in a given volume V s at some
instant, instead of asking what volume N s molecules happen to occupy
at a given instant, as we did here. The results will of course turn out
the same, as will be shown in Chapter 23.
Brownian Motion
The fluctuating motion of a small particle in a fluid, caused by the
thermal fluctuations of pressure on the particle, is called Brownian
motion. We have already seen [in Eq. (153)] that the mean square of
FLUCTUATIONS 129
each of the velocity components of such motion is proportional to T
and inversely proportional to the mass of the particle. The mean
square of each of the position coordinates of the particle is not as
simple to work out for the unbound particle as it was for the displace
ment of the mass on a spring, discussed in the preceding section.
In the case of the mass on the end of the spring, the displacement
x from equilibrium is confined by the restoring force, the maximum
displacement is determined by the energy possessed by the oscillator,
and we can measure a meansquare displacement from equilibrium,
<x 2 >, by averaging the value of x 2 over any relatively long interval
of time (the longer the interval, the more accurate the result, of
course). But the x component of displacement of a free particle in a
fluid is not so limited; the only forces acting on the particle (if we
can neglect the force of gravity) are the fluctuations of pressure of
the fluid, causing the Brownian motion, and the viscous force of the
fluid, which tends to retard the particle's motion. If we measure the
x component of the particle's position (setting the initial position at
the origin) we shall find that, although the direction of motion often
reverses, the particle tends to drift away from the origin as time goes
on, and eventually it will traverse the whole volume of fluid, just as
any molecule of the fluid does. If we allow enough time, the particle
is eventually likely to be anywhere in the volume. This is in corre
spondence with the Maxwell Bo ltzmann distribution; the potential en
ergy does not depend on x (neglecting gravity) so the probability den
sity f is independent of x; any value of x is equally likely in the end.
Bat this was not the problem at present. We assumed that the par
ticle under observation started at x = 0; it certainly isn't likely to
drift far from the origin right away. Of course the average value of x
is zero, since the particle is as likely to drift to the left as to the
right. But the expected value of x 2 must increase somehow with time.
At t = the particle is certainly at the origin; as time goes on the
particle may drift farther and farther away from x = 0, in either di
rection. We wish to compute the expected value of x 2 as a function of
time or, better still, to find the probability density of finding the par
ticle at x after time t. Note that this probability is a conditional
probability density; it is the probability of finding the x component of
the position of the particle to be x at time t if the particle was at the
origin at t = 0.
Random Walk
We can obtain a better insight into this problem if we consider the
randomwalk problem of Eqs. (119) and (1110). A crude model of
onedimensional Brownian motion can be constructed as follows. Sup
pose a particle moves along a line with a constant speed v. At the end
of each successive time interval i it may change its direction of mo
130
KINETIC THEORY
tion or not, the two alternatives being equally likely (p = 1/2) and dis
tributed at random. After N intervals (i.e., after time Nr) the chance
that the particle is displaced an amount x n = (2n  N)vt from its initial
position is then the chance that during n of the N intervals it went to
the right and during the other (N n) intervals it went to the left; it
covered a distance vr in each interval. According to Eq. (115) this
probability is
N
P n (N) = N!(l/2) i 7n!(Nn)!
(157)
since p = 1/2 for this case.
The expected value of x n and its variance are then obtained from
Eqs. (119) and (1110) (for p = 1/2):
N
<x>
L
n =
(2nN)vrP n (N) =
N
<x 2 >= 2Z < 2n
n =
N) 2 (vt ) 2 P n (N) = N(vt f
(158)
The expected value of the displacement is zero, since the particle is
as likely to go in one direction as in the other. Its tendency to stray
from the origin is measured by <x 2 >, which increases linearly with
the number of time intervals N, and thus increases linearly with time.
If the particle, once started, continued always in the same direction
(p = or 1) the value of <x 2 > would be (Nvr) 2 , increasing quadratic
ally with time But with the irregular, to and fro motion of the random
walk, <x 2 > increases only linearly with time. Figure 153 is a plot
Fig. 153. Displacement for random walk. At each dot the
"walker" flipped a coin to decide whether to
step forward or backward.
FLUCTUATIONS 131
of x as a function of t for a random walk as described here. We note
the irregular character of the motion and the tendency to drift away
from x = 0. Compare it with the curves of Fig. 151, and also with
152, for a mass with restoring force.
The limiting case of N very large and r very small is the case
most nearly corresponding to Brownian motion. This limiting form
was calculated at the end of Chapter 11. There we found that the prob
ability distribution for displacement of the particle after N steps re
duced, in the limit of N large, to the normal distribution of Eq. (1117),
The variance, in this case, as we just saw, is a 2 = (vr) 2 N (for p = 1/2)
and the mean value of x is X = 0. Since the time required for the N
steps is t = tN we can write the conditional probability density (that
the particle will be at x at time t if it was at x = at t = 0) as
F(x) = [l/(47TDt) l/2 ] e~ x2 /4Dt. D =  v 2 r (159)
so that the probability that the particle is between x and x + dx at
time t is F(x) dx. We see that the "spread" of the distribution in
creases as t increases, as the particle drifts away from its initial
position. The mean square value of x is
<x 2 > = (J 2 = (vt) 2 N= 2Dt
which increases linearly with time. Thus the question of the depend
ence of <x 2 > on time, raised earlier in this section, is answered to
the effect that <x 2 > is proportional to t. The value of the propor
tionality constant 2D for the actual Brownian motion of a particle in
a fluid must now be determined.
The Langevin Equation
To determine the value of constant D for a particle in a fluid we
must study its equation of motion. As before, we study it in a single
dimension first. The x component of the force on the particle can be
separated into two parts, an average effect of the surrounding fluid
plus a fluctuating part, caused by the pressure fluctuations of thermal
motion of the fluid. The average effect of the fluid on the particle is a
frictional force, caused by the fluid's viscosity. If the velocity of the
particle in the x direction is x, this average frictional force has an
x component equal to M/3x, opposing the particle's motion, where the
mechanical resistance to motion, M/3, in a fluid of viscosity 77, on a
spherical particle of radius a is M/3 = 67ra?] (Stoke' s law). The fluc
tuating component of the force on the particle can simply be written
as MA(t) (we write these functions with a factor M, the mass of the
particle, so that M can be divided out in the resulting equation).
The equation of motion for the x component of the particle's posi
tion can thus be written as
132 KINETIC THEORY
Mx = M/3x + MA(t) (1510)
which is known as Langevin' s equation. We note that /3 has the di
mensions of reciprocal time. Multiplying the equation by x/M and us
ing the identities
i«w and xX= l5
xx = 77 37 (x 2 ) and xx =  ^ (x 2 )  (x) 2
we have
1 J?L (X 2) = oA
2 dt 2 iX ; P dt
^3TF(x 2 )= /3^(x 2 )+(x) 2 + xA(t)
This is an equation for one particular particle. If we had many iden
tical particles in the fluid (or if we performed a sequence of similar
observations on one particle) each particle would have different values
of x and x at the end of a given time t, because of the effects of the
random force A(t).
Suppose we average the effects of the fluctuations by averaging the
terms of Eq. (1510) over all similar particles. The term xA(t) will
average out because both <x> and <A> are zero and the fluctuations
of x and A are independent; the average value of x 2 , however, car
ries with it the mean effects of the fluctuating force A(t). We showed
in Eq. (153) that for a particle in thermal equilibrium at temperature
T, its meansquare velocity component <x 2 > is equal to kT/M.
If we now symbolize the meansquare displacement <x 2 > as s(t),
the average of the equation of motion written above turns out to be
 s = (kT/M)  /3s
The solution of this equation is s = (2kT/M/3)  Ce"^. The transient
exponential soon drops out, leaving for the steadystate solution
s = (2kT/M/3) and thus
s = <x 2 > = (2kT/M/3)t (1511)
This result answers the question raised at the end of the previous sec
tion; the constant D = (1/2)(v 2 t), used there now turns out to be
kT/M/3 and, for a spherical particle of radius a in a fluid of viscosity
77, constant D is equal to kT/67ra77 from Stoke 's law.
The innocuous looking result shown in Eq. (1511) enabled Perrin
and others first to measure Avogadro's number N and thus, in a
sense, first to make contact with atomic dimensions. They were able
to measure N in terms of purely macroscopic constants, plus obser
FLUCTUATIONS 133
vations of Brownian motion. A spherical particle was used, of known
radius a, so that Stoke 's law applied. The viscosity of the fluid in
which the particle was immersed was measured as well as the tem
perature T of the fluid. The value of the gas constant R was known
but at the time neither the value of the Boltzmann constant k nor the
value of N = R/k was known. The x coordinates of the particle in
the fluid were measured at the ends of successive intervals of time of
length t; Xq at t = 0, x x at t, x 2 at 2t, and so on, and the average of
the set of values (x n+ j  x n ) 2 was computed, which is equivalent to
the <x 2 > of Eq. (1511).
By making the measurements for several different values of the
time interval t, it was verified that <x 2 > does indeed equal 2Dt,
and the value of D was determined. The value of Avogadro's number,
N = R/k = RT/67ra?7D
can thus be computed. By this method a value of N was obtained
which checks within about 5 parts in a thousand with values later ob
tained by more direct methods. Of course very small spheres had to
be used, to make <x 2 > as large as possible, and careful observations
with a microscope were made to determine the successive x n 's.
Perrin used spheres of radius 2 x 10 5 cm and time intervals from a
few seconds to a minute or more.
The Fokker Planck Equations
Brownian motion is simply the fine details of the process of diffu
sion. If there were initially a concentration of particles in one region
of the fluid, as time went on these particles would diffuse, by Brown
ian motion, to all parts of the fluid. The diffusion constant for a par
ticle in a fluid is D = kT/M£, which is to be compared with Eq.
(148) for the D for a molecule; in the molecular case /3 is evidently
equal to l/t c , whereas for a larger, spherical particle j3= Qii3.ri/M.
The mean concentration of the diffusing particles must satisfy a dif
fusion equation of the type given in Eq. (1410).
This means that there is a close connection between the results of
this chapter and those of the section on diffusion. Whether the diffusing
entity undergoing Brownian motion is a molecule of the gas or a dust
particle in the gas, the probability density of its presence at the point
in space given by the vector r at time t, if it starts at r at t = 0, is
given by the threedimensional generalization of Eq. (159),
f r (r,t) = (47rDt) 3/2 exp ( F " r ° 2 ]; D = kT/M/3 (1512)
which is a solution of the diffusion equation (1410). The value of /3
134 KINETIC THEORY
appropriate for the particle under study must be used, of course.
The distribution function for diffusion by Brownian motion of Eq.
(1512) and the diffusion equation (1410) that it satisfies can thus be
derived by the methods of the previous chapter or else by those of
this chapter. For example, it is possible to generalize the Langevin
equation (1510) and manipulate it to obtain the diffusion equation.
Also, by either method, it can be shown that when an external force
F acts on the diffusing particle (such as the force of gravity), the dif
fusion equation has the more general form
9f r /9t = div [D grad f r  (F/M/3)f r ] (1513)
When f r is the density of diffusing substance (molecules or heat,
for example), Eq. (1513), or its simple version (1410) for F = 0, is
called the diffusion equation. When f r is the distribution function for
a particle undergoing Brownian motion and the equation is considered
to be a first approximation to a generalized Langevin equation, then
Eq. (1513) is called a Fokker Planck equation. The solutions behave
the same in either case, of course. The solution of (1513) for F =
and for the particle starting at r = r when t = is Eq. (1512). From
it can be derived all the characteristics of Brownian motion in regard
to the possible position of the particle at time t.
A Fokker Planck equation can also be obtained for the distribution
in momentum, f p (p,t) of the particle. It is
Bfp/at = div p (MkT grad p f p + f p p) (1514)
For a particle that is started at t = with a momentum p = p , the
solution of this equation, which is the probability density of the parti
cle in momentum space, is
^£t' 2
f D (p,t) = [27iMkT(l  e" 2 ^)] 3/2 exp
P e
2MkT(le" 2/3t )
(1515)
This interesting solution shows that the expected momentum of the
particle at time t is p e~£t [compare with the discussion of Eq.
(1117)] , which is the momentum of a particle started with a momen
tum p and subjected to a frictional retarding force pp. As time goes
on, the effect of the fluctuations "spreads out" the distribution in mo
mentum; the variance of the momentum (i.e., its meansquare devia
tion from p e"~P*) is kT(l  e  2/3t) ? starting as zero when t = 0, when
we are certain that the particle's momentum is p , and approaching
asymptotically the value kT, which is typical of the Maxwell distribu
tion. Thus Eq. (1515) shows how an originally nonequilibrium distri
bution for a particle (or a molecule) in a fluid can change with time
into the Maxwell distribution typical of an equilibrium state. Constant
FLUCTUATIONS 135
)3, which equals Qtislti/M for a spherical particle or l/t c for a mole
cule in a gas, is thus equal to the reciprocal of the relaxation time for
the distribution, which relates directly to the discussion of Eq. (142).
Of course the most general distribution function would be f(r,p,t),
giving the particle's distribution in both position and momentum at
time t after initial observation. The equation for this f is, not sur
prisingly, closely related to the Boltzmann equation (132). It can be
shown to be
i + J6L . grad r f + F • grad p f = 9 div p [MkT grad p f + pf ]
(1516)
The derivation of this equation, particularly of the righthand side of
it, involves a generalization of the arguments used in deriving Eq.
(1511). This righthand side is another approximation to the collision
term Q of Eq. (132).
Ill
STATISTICAL
MECHANICS
Ensembles
and
Distribution
Functions
It is now time to introduce the final generalization, to present a
theoretical model that includes all the special cases we have been
considering heretofore. If we had been expounding our subject with
mathematical logic we would have started at this point, presenting
first the most general assumptions and definitions as postulates,
working out the special cases as theorems following from the
postulates, and only at the end demonstrating that the predictions,
implicit in the theorems, actually do correspond to the "real
world," as measured by experiment. We have not followed this
procedure, for several reasons.
In the first place, most people find it easier to understand a new
subject, particularly one as complex as statistical physics, by pro
gressing from the particular to the general, from the familiar to the
abstract.
A more important reason, however, is that physics itself has
developed in a nonlogical way. Experiments first provide us with
data on many particular cases, at first logically unconnected with
each other, which have to be learned as a set of disparate facts.
Then it is found that a group of these facts can be considered
to be special cases of a "theory/ ' an assumed relationship be
tween defined quantities (energy, entropy, and the like) which
will reproduce the experimental facts when the theory is appropri
ately specialized. This theory suggests more experiments, which
may force modifications of the theory and may suggest further
generalizations until, finally, someone shows that the whole sub
ject can be "understood' ' as the logical consequences of a few
basic postulates.
At this point the subject comes to resemble a branch of mathe
matics, with its postulates and its theorems logically deduced there
from. But the similarity is superficial, for in physics the experimental
facts are basic and the theoretical structure is erected to make it
139
140 STATISTICAL MECHANICS
easier to "understand" the facts and to suggest ways of obtaining new
facts. A logically connected theory turns out to be more convenient to
remember than are vast arrays of unconnected data. This convenience,
however, should not persuade us to accord the theory more validity
than should inhere in a mnemonic device. We must not expect, for ex
ample, that the postulates and definitions should somehow be "the
most reasonable" or "the logically inevitable" ones. They have been
chosen for the simple, utilitarian reason that a logical structure
reared on them can be made to correspond to the experimental facts.
Thus the presentation of a branch of physics in "logical" form tends
to exaggerate the importance and inevitability of the theoretical as
sumptions, and to make us forget that the experimental data are the
only truly stable parts of the whole.
This danger, of ascribing a false sense of inevitability to the the
ory, is somewhat greater with statistical physics than with other
branches of classical physics, because the connection between experi
ment and basic theory is here more indirect than usual. In classical
mechanics the experimental verification of Newton's laws can be
fairly direct; and the relationship between Faraday's and Ampere's
experiments and Maxwell's equations of electromagnetic theory is
clearcut. In thermodynamics, the experiments of Rumford, relating
work and heat, bear a direct relationship to the first law, but the ex
perimental verification of the second law is indirect and negative.
Furthermore, the more accurate "proofs" that the MaxwellBoltz
mann distribution is valid for molecules in a gas, are experimentally
circuitous. And finally, as we shall see later, there is no experiment,
analogous to those of Faraday or Ampere, which directly verifies any
of the basic assumptions of statistical mechanics; their validity must
be proved piecemeal and infer enti ally. In the end, of course, the
proofs are convincing from their very number and breadth of applica
tion.
However we have now reached a point in our exposition where the
basic theory must be presented, and it is necessary to follow the pat
tern of mathematical logic for a time. Our definitions and postulates
are bound to sound arbitrary until we have completed the demonstra
tion that the theory does indeed correspond to a wide variety of ob
served facts. But we must keep in mind that they have been chosen
solely to obtain this correspondence with observation, not because
they "sound reasonable" or satisfy some philosophical "first prin
ciples."
Distribution Functions in Phase Space
In Chapters 13 and 15 we discussed distribution functions for mole
cules, and also for multimolecular particles in a gas. In statistical
ENSEMBLES AND DISTRIBUTION FUNCTIONS 141
mechanics we carry this generalization to its logical conclusion, and
deal with distribution functions for complete thermodynamic systems.
A particular microstate of such a system (a gas of N particles, for
example) can be specified by choosing values for the 6N position and
momentum coordinates; the distribution function is the probability
density that the system has these coordinate values. Geometrically
speaking, an elementary region in this 6Ndimensional phase space
represents a microstate of the system; the point representing the sys
tem passes through all the microstates allowed by its thermodynamic
state; the fraction of time spent by the system point in a particular
region of phase space is proportional to the distribution function cor
responding to the thermodynamic state. In other words, a choice of a
particular thermodynamic state (a macrostate) is equivalent to a
choice of a particular distribution function, and vice versa. The task
of statistical mechanics is to devise methods for finding distribution
functions which correspond to specific macrostates.
According to classical mechanics, the distribution function for a
system with = 3N degrees of freedom is
f(q,p) =f(qi,q 2 >..., q0,Pi,p 2 ,...,P0)
where the q's are the coordinates and the p's the conjugate momenta
[see Eq.(139)] which specify the configuration of the system as a
whole. Then f(q,p) dVq dVp [where dVq = hidq 1 h2dq2***h^ ) dq0 and
dVp = (dp 1 /hi)(dp2/h 2 )(dp0/h ( ^)] is the probability that the system
point is within the element of phase space dVq dVp at position qi,...,P0,
at any arbitrarily chosen instant of time.
More generally, the distribution function represents the probability
density, not for one system, but for a collection of similar systems.
Imagine a large number of identical systems, all in the same thermo
dynamic state but, of course, each of them in different possible micro
states. This collection of sample systems is called an ensemble of
systems, the ensemble corresponding to the specified macrostate.
Different ensembles, representing different thermodynamic states,
have different populations of microstates. The distribution function
for the macrostate measures the relative number of systems in the
ensemble which are in a given microstate at any instant. Thus it is a
generalization of the distribution function of Eq. (1310), which was
for an ensemble of molecules.
Each system in the ensemble has coordinates q^ and mo
menta pi, and a Hamiltonian function H(q,p), which is the total energy
of the system, expressed in terms of the q's and p's. The values of
these q's and p's at a given instant determine the position of its
system point in phase space. The motion of the system point in phase
space is determined by the equations of motion of the system. These
142 STATISTICAL MECHANICS
can be expressed in many forms, each of which are discussed in books
on classical dynamics. The form which is most useful for us at pres
ent is the set of Hamilton's equations,
qi = 3H/api; pi= BH/aqi, i = 1,2,. ..,0 (161)
the first set relating the velocities qi to the momenta and the second
set relating the force components to the rates of change of the mo
menta (the "velocity components" in momentum space).
The ensemble of systems can thus be represented as a swarm of
system points in phase space, each point moving in accordance with
Eqs. (161); the velocity of the point in phase space is proportional
to the gradient of H. The density of points in any region of phase
space is proportional to the value of the distribution function f(q,p)
in that region.
Liouville's Theorem
If the thermodynamic state is not an equilibrium state, f will be a
function of time. If the state is an equilibrium state, the density of
system points in any specified region of phase space will be constant;
as many system points will enter the region per unit of time as will
leave it. The swarm of system points has some similarity to the
swarm of particles in a gas. The differences are important, how
ever. The system points are moving in a 20 dimensional phase
space, not real space; also the system points do not collide. In fact
the system points do not interact at all, for each system point repre
sents a different system of the ensemble, and the separate systems
cannot interact since they are but samples in an imaginary array of
systems, assembled to represent a particular macrostate. Each in
dividual system point, for example, may represent a whole gas of N
atoms, or a crystal lattice, depending on the situation the ensemble
has been chosen to represent. There can be no physical interaction
between the individual sample systems.
This means that the Boltzmann equation for the change of f with
time, the generalization of Eq. (132) to the ensemble, has no colli
sion term Q. The equation,
% + L£p*> + L£pti° (16  2)
i = 1 i = 1
is simply the equation of continuity in phase space, and represents
the fact that, as the swarm of system points moves about in phase
space, no system point either appears or disappears.
ENSEMBLES AND DISTRIBUTION FUNCTIONS 143
Since each system in the ensemble obeys Hamilton's equations
(161), this equation of continuity becomes
at
Z_i L S( li\ a Pi/ a Pi\ 8c U/_
=
and since
8H df_ = _d_ L m \ _ f a 2 H
api aqi 9qi\3Pi/ SQiSPi
etc., we have
i = 1 i = 1
where df/dt is the change in the distribution function f in a coordi
nate system which moves with the system points. Because of the re
lationship between q and p, p and q, inherent in Hamilton's equa
tions, the density of system points near a given system point of the
ensemble remains constant as the swarm moves about. If, at t = 0,
the swarm has a high density in a localized region of phase space,
this concentration of system points moves about as time goes on but
it does not disperse; it keeps its original high density. This result is
known as Liouville's theorem.
We can use Liouville's theorem to devise distribution functions
which are independent of time, i.e., which represent equilibrium
macrostates. For example, if f had the same value everywhere in
phase space, it would be independent of time; as a part of the swarm
moved away from a given region of phase space a different part of
the swarm would move in and, since all parts of the swarm have (and
keep) the same density, the density in a given region would not change,
We can also devise less trivial stationary distributions, for the path
traversed by any one system point does not cover all of phase space;
it confines itself to the hypersurface on which the Hamiltonian func
tion H(q,p) is constant; an isolated system cannot change its total
144 STATISTICAL MECHANICS
energy. Therefore if the distribution function is the same for all re
gions of phase space for which H is the same (i.e., if f is a function
of H alone) the density of system points in a given region cannot
change as the points move along their constant H paths.
We shall deal with several different types of distribution functions,
corresponding to different specifications regarding the thermody
namic state of the system. The simplest one is for f to be zero
everywhere in phase space except on the hypersurface corresponding
to H(q,p) = E, a constant; the ensemble corresponding to this is called
a microcanonical ensemble. A more useful case is for f to be pro
portional to exp[H(q,p)/kT], corresponding to what is called the
canonical ensemble. Other ensembles, with f's which are more
complicated functions of H, will also prove to be useful. But, in order
for any of them to represent actual thermodynamic macrostates, we
must assume a relationship between the distribution function f for
an ensemble and the corresponding thermodynamic properties of the
macrostate which the ensemble is supposed to represent. The ap
propriate relationship turns out to be between the entropy of the
macrostate and the distribution function of the corresponding ensemble.
Quantum States and Phase Space
Before we state the basic postulate of statistical mechanics, re
lating entropy and ensembles, we should discuss the modifications
which quantum theory makes in our definitions. In some respects the
change is in the direction of simplification, the summation over de
numerable quantum numbers being substituted for integration over
continuous coordinates in phase space. Instead of Hamilton's equa
tions (161), there is a Shrodinger equation for a wave function
^(Qi 3 ci2,..,q(/)) and an allowed value E of energy of the system, both
of which depend on the 6 quantum numbers v lf v 2 ,...,v§, which desig
nate the quantum state of the system.
For example, if the system consists of N particles in the simple
crystal model of Eq. (132), the classical Hamiltonian for the whole
system can be written as
(b
H(q,p) = Z [l/2m)pf + (mw72)qf] (0 = 3N) (164)
j =1
where q 3i _ 2 = x i? q 3i _ 1 = y i5 q 3i = z it p 3i _ 2 = Pxi> etc  Hamil
ton's equations become
qi  (l/m)pi; Pi = mw 2 qi (165)
and Schrodinger's equation for the system is H^ = E\£, where each
ENSEMBLES AND DISTRIBUTION FUNCTIONS 145
Pi in the H is changed to (h/i)(8/3qi). For (164) it is
i  1
where Planck's constant h is equal to 27rh. Solution of this equation,
subject to the requirement that ^ is finite everywhere, results in the
following allowed values of the energy E,
E, . = Y[ hco^i+l] (167)
The distribution function for an equilibrium state is a function of
E^ij'v^d) an< ^ ^ nus is a f unc ti° n of the <fi quantum numbers ^1,1^2,...,
VAy, instead of being a function of H(q,p) and thus a function of the 20
continuous variables qi,q2,...,q0>Pi>...>P0, as it was in classical
mechanics. Function f^i,...,^) is the probability that the system is
in the quantum state characterized by the quantum numbers Vi,...,V(p;
as contrasted with the probability f(qi,...,P0) dVq dVp for phase
space. These statements apply to any system, not just to the simple
crystal model. The quantum state for any system with $ degrees of
freedom, no matter what conservative forces its particles are sub
jected to, is characterized by quantum numbers ^...,^a. To
simplify notation we shall often write the single symbol v instead of
the cf) individual numbers Vi 9 .*.,V(h, just as we write q for qi,...,q0,
etc. Thus i v is the probability that the system is in the quantum
state v = vi f ...,V(j). The sum J^fv over all allowed states of the sys
^ v
tern must be unity, of course.
We thus have two alternative ways of expressing the microstates
of the system, and thus of writing the distribution function. The quan
tum way, saying that each quantum state of the system is a separate
and distinct microstate, is the correct way, but it sometimes leads to
computational difficulties. The classical way, of representing a
microstate as a region of phase space, is only an approximate repre
sentation, good for large energies; but when valid it is often easier to
handle mathematically. The quantitative relationship between these '
two ways is obtained by working out the volume of phase space "oc
cupied" by one quantum state of the system.
The connection between the classical coordinates q^ and momenta
Pi and the quantum state is provided by the Heisenberg uncertainty
146 STATISTICAL MECHANICS
principle, Aq^ * Apj ^ h. A restatement of this is that, in the phase
space of one degree of freedom, one quantum state occupies a "vol
ume" Aq i Ap i equal to h. For example, the one dimensional harmonic
oscillator has a Hamiltonian Hi = (l/2m)pi + (mu; 2 /2)qi. When in the
quantum state v\, with energy hco(^ + 1/2) = (hw/27r)(i>i + 1/2), its
phase space orbit is an ellipse in phase space, with semiminor axis
q m = [(h/7imco)(^i + 1/2)] 1/2 along q^ and semimajor axis p m =
[(hma;/7r)(^ + 1/2)] V2 , which encloses an area
Ad>i.) =7rp m q m =h^i + 2)
The area between successive ellipses, for successive values of v^ is
the area "occupied" by one quantum state. We see that A(i^ + 1) 
A(i^) = h, as stated above.
Thus a volume element dq^ dpi corresponds, on the average, to
(dqi dpi/h) quantum states. Similarly, for the whole system, with
degrees of freedom, the volume element dVq dVp = dq^.dp^ will
correspond, on the average, to (dVq dVp/h^) quantum states. Thus
the correspondence between volume of phase space and number of
quantum states is
No. of microstates = (l/h0)(vol. of phase space) (168)
when the system has degrees of freedom.
When the volume of phase space occupied by the swarm of sys
tem points, representing a particular ensemble, is very large com
pared to h0, the classical representation, in terms of the continuous
variables qi,...,?^ can be safely used. But when the volume occupied
by the swarm is not large compared to h^, the classical representa
tion is not likely to be valid and the quantum representation is needed
[see Eq. (198) et seq.] .
Entropy
and Ensembles
As pointed out in the preceding chapter, we are presenting statis
tical mechanics in "logical" order, with definitions and basic postu
lates first, theorems and connections with experiment later. The last
chapter was devoted to definitions. Each thermodynamic macrostate
of a system may be visualized as an ensemble of systems in a variety
of microstates, or may be represented quantitatively in terms of a
distribution function, which is the probability i v that a system chosen
from the ensemble is in the quantum state v = v ly v 2 ,...,v ( k or is the
probability density f(q,p) that the system point has the coordinates
( 1i> c 12>"'jP0 in Phase space, if the macrostate is such that classical
mechanics is valid. In this chapter we shall introduce the essential
postulates.
Entropy and Information
The basic postulate, relating the distribution function i v to the
thermodynamic properties of the macrostate which the ensemble rep
resents, was first stated by Boltzmann and restated in more general
form by Planck. In the form appropriate for our present discussion
it relates the entropy S of the system to the distribution function i v
by the equation
S = k£fy ln(f^); £fi/ = l (171)
v v
where k is the Boltzmann constant and where the summation is over
all the quantum states present in the ensemble (i.e., for which i v
differs from zero).
This formula satisfies our earlier statements that S is a measure
of the degree of disorder of the system [see discussion preceding
Eq. (614)]. A system that is certainly in its single, lowest quantum
state is one in perfect order, so its entropy should be zero. Such a
147
148 STATISTICAL MECHANICS
system would have the i v for the lowest quantum state equal to unity
and all the other f's would be zero. Since ln(l) = and x ln(x) —
as x — 0, the sum on the righthand side of Eq. (171) is zero for this
case. On the other hand, a disorderly system would be likely to be in
any of a number of different quantum states; the larger the number of
states it might occupy the greater the disorder. If i v = 1/N for N
different microstates (label them v = 1,2,...,N) and i v is zero for all
other states then
N
S = k S (1/N) ln(l/N) = k In N
v = 1
which increases as N increases. Thus Eq. (171) satisfies our pre
conceptions of the way entropy should behave. It also provides an op
portunity to be more exact in regard to the measurement of disorder.
Disorder, in the sense we have been using it, implies a lack of in
formation regarding the exact state of the system. A disordered sys
tem is one about which we lack complete information. Equation (171)
is the starting point for Shannon's development of information theory.
It will be useful to sketch a part of this development, for it will cast
further light on the meaning of entropy, as postulated in Eq. (171).
Information Theory
A gasoline gauge, with a pointer and scale, gives us more informa
tion about the state of the gasoline tank of an automobile than does a
red light, which lights if the tank is nearly empty. How much more?
Information comes to us in messages and to convey information each
message must tell us something new, i.e., something not completely
expected. Quantitatively, if there are N possible messages that could
be received, and if the chance that the ith one will be sent is fj, then
the information I that would be gained if message i were received
must be a function l(fi), which increases as 1/fi increases. The less
likely the message, the greater the information conveyed if the mes
sage is sent.
We can soon determine what function l(fj) must be, for we require
that information be additive; if two messages are received and if the
messages are independent, then the information gained should be the
sum of the I's for each individual message. If the probability of
message i be fj and that for j be fj then, if the two are independent,
Eq. (113) requires that the probability that both messages happen to
be sent is f ^f j . The additive requirement for information then re
quires that
I(f if j ) = I(f i) +I(fj)
ENTROPY AND ENSEMBLES 149
and this, in turn, requires that function I be a logarithmic function
of f,
I(fi) = Clnfi
where C is a constant. This is the basic definition of information
theory. Since ^ fi ^ 1, I is positive and increases as 1/fi increases,
as required.
The definition satisfies our preconceptions of how information be
haves. For example, if we receive a message that is completely ex
pected (i.e., its a priori probability is unity) we receive no informa
tion and I is zero. The less likely is the message (the smaller is fi)
the greater the amount of information conveyed if it does get sent.
The chance that the warning light of the gasoline gauge is off (showing
that the tank is not nearly empty) is 0.9, say, so the information con
veyed by the fact that the light is not lit is a small amount, equal to
C In 0.9  0.1C. On the other hand if the gauge has a pointer and five
marks, each of which represents an equally likely state of the tank,
then the information conveyed by a glance at the gauge is C In 5  1.6C,
roughly 16 times the information conveyed by the unlit warning light
(the information conveyed by a lit warning light, however, is C In 10 
2.3C, a still larger amount).
To see how these definitions relate to our discussion of information
and disordered systems, let us return to an ensemble, corresponding
to some thermodynamic state, with its distribution function ip. If we
wish to find out exactly what microstate the system happens to be in
at any instant, we would subject it to detailed measurement designed
to tell us. The results of the measurement would be a kind of mes
sage to us, giving us information. If the measurements happened to
show that the system is in microstate v, the information gained by
the measurement would be C lnf^, for i v is the probability that the
system would happen to be in state v. We of course cannot make a
detailed enough examination to determine exactly what microstate a
complicated system should happen to be in, nor would we wish to do so
even if we could. But we can use the expected amount of information
we would obtain, if we made the examination, as a measure of our
present lack of knowledge of the system, i.e., of the system's dis
order.
The expected amount of information we would obtain if we did ex
amine the system in detail is the weighted mean of C lnfj, over all
quantum states v in the ensemble, the weighting factor being the
probability i v of receiving the message that the system is in state
v. This is the sum C£fy lnfy = (C/k)S, according to Eq. (171).
v
Thus the entropy S is proportional to our lack of detailed informa
tion regarding the system when it is in the thermodynamic state
150 STATISTICAL MECHANICS
corresponding to the distribution function i v . Here again Eq. (171)
corresponds to our preconceptions regarding the entropy S of a
system.
Entropy for Equilibrium States
But we do not wish to use postulate (171) to compute the entropy
of a thermodynamic state when we know its distribution function; we
wish to use Eq. (171) to find the distribution function for specified
thermodynamic states, particularly those for equilibrium states. In
order to do this we utilize a form of the second law. We noted in our
initial discussion of entropy [see Eq. (65)] that in an isolated system
S tends to increase until, at equilibrium, it is as large as it can be,
subject to the restrictions on the system. If the sum of Eq. (171) is
to correspond to the entropy, defined by the second law, it too must
be a maximum, subject to restrictions, for an equilibrium state.
These requirements should serve to determine the form of the dis
tribution function, just as the wave equation, plus boundary condi
tions, determines the form of a vibrating string.
To show how this works, suppose we at first impose no restric
tions on f^, except that Yj^p = 1 an< ^ that the number of microstates
v
in the ensemble represented by f v is finite (so that the quantum num
ber v can take on the values 1,2,...,W, where W is a finite integer).
Then our problem is to determine the value of each i v so that
W W
S = k Yj ^p Infy is maximum, subject to Yj ^p = 1 (172)
p = l v=\
This is a fairly simple problem in the calculus of variations, which
can be solved by the use of Lagrange multipliers. But to show how
Lagrange multipliers work, we shall first solve the problem by
straightforward calculus.
The requirement that S^ = 1 means that only Wl of the f's
can be varied independently. One of the f's, for example fw, depends
W 1
on the others through the equation f^y = 1  Yj *v • Now S is a
p = 1
symmetric function of all the f s, so we can write it S(f!,f 2 ,...,fw)>
where we can substitute for f\v in terms of the others. In order that
S be maximum we should have the partial derivative of S with re
spect to each independent f be zero. Taking into account the fact that
f w depends on all the other f's, these equations become
ENTROPY AND ENSEMBLES
151
(ds/du) + (afw/afi)(3S/afw) = tes/af x )  (as/af w ) = o
(as/ay  (8S/af w )=o (17 _ 3)
(as/8f w _!)  (as/af w ) = o
The values of the f's which satisfy these equations, plus the
equation J^iv = l, are those for which Eq. (172) is satisfied. For
these values of the f's, the partial derivative 8S/af\y will have some
value; call it a . Then we can write Eqs. (173) in the more sym
metric form
©♦■• K)*""°
(*)♦
However this is just the set of equations we would have obtained if,
instead of the requirement that Sff^'^fw) De maximum, subject to
Yjiv = 1 of Eq. (172), we had instead used the requirement that
W
S(f 1 ,...,fw) + a o S iv be maximum,
v = l
W
q determined so that 2 ^v ~ 1
(174)
Constant a is a Lagrange multiplier.
Let us now solve the set of equations (174), inserting Eq. (171)
for S. We set each of the partials of S + a Q J^i v equal to zero. For
example, the partial with respect to U is
=
dk
W
W
«o £_,**>  k Zj tvhiip
a klnf K k
or
t K = exp[(a /k)  1]
152 STATISTICAL MECHANICS
The solution indicates that all the f's are equal, since neither a
nor k depend on k. The determination of the value of a , and thus
of the magnitude of i Vi comes from the requirement Yj^v ~ l'» *v ~
(1/W), so that
W
s =  k H (i) ln (w) =klnW (17 " 5)
i/ = 1
For a system restricted to a finite number W of microstates, and
with no other restrictions, the state of maximum entropy is that for
which the system is equally likely to be in any of the W microstates,
and the corresponding maximal value of the entropy is k times the
natural logarithm of the number W (which is sometimes called the
statistical weight of the equilibrium macrostate).
Application to a Perfect Gas
To show how much is inherent in these abstract sounding results,
we apply them to a gas of N point particles, confined in a container
of volume V. To say that the system is confined to a finite number of
quantum states is equivalent, classically, to saying that the system
point is confined to a finite volume in phase space. In fact, from Eq.
(166), the volume of phase space fi, within which the system point is
to be found, is fi = h^N^, where = 3N is the number of degrees of
freedom of the gas of N particles. And, from Eq. (175) we see that
the system point is equally likely to be anywhere within this volume
Q. Thus, classically the analogue of Eq. (175) is
f(q,p) = I/O; S = k ln(S2/h3N) (176)
As long as the volume of the container V is considerably larger
than atomic dimensions, Q is likely to be considerably larger than
h3N, so the classical description, in terms of phase space, is valid
[see the discussion following Eq. (166)].
The volume ft can be computed by integrating dVq dVp over the
region allowed to the system. Since each particle is confined to the
volume V, the integration over the position coordinates is
/— /dV q = /.../dx! d Yl dz x — dx N dy N dz N = V N (177)
The integration of dVp will be discussed in the next chapter; here
we shall simply write it as ftp. Therefore,
ENTROPY AND ENSEMBLES 153
$2 = V N ft p and S = Nk In V + k ln(ftp/h 3N ) (178)
Comparison with Eq. (65) shows that the entropy of a perfect gas
does indeed have a term Nk In V(Nk = nR).
Moreover in this case of uniform distribution within £2, the mean
energy of the gas, U = £fyEy, wi ^ De given by the integral
u =i/.../dv q /.../ H dv p = iffn dv p
where
N
H = (l//2m) £ (Pxi + Pvi +Pzi)
3=1
is the total energy of the perfect gas. Thus U is a function of ftp, as
well as of m and N and the shape of the volume in phase space,
within which the ensemble is confined; we can emphasize this by
writing it as U(ftp). Note that this is so only when H is independent
of the q's. However, from Eq. (178) we have
ftp = (h3/V)N e S/k so U = U(h3N V N e S/k)
Thus the formalism of Eq. (174) has enabled us to determine
something about the dependence of the internal energy on V and S for
a system with H independent of the q's. We do not know the exact
form of the dependence, but we do know that it is via the product
yN eS/k^ From this one fact we can derive the equation of state for
the perfect gas. We first refer to Eqs. (81). If the postulates (171)
and (174) are to correspond to experiment, the partials of the function
U(h3NyN e S/k) W ith respect to S and V must equal T and P, re
spectively. But
(§HS> SA (f£ P ) 
Thus, if the first partial is to equal the thermodynamic temperature
T and minus the second partial is the pressure P for any system
with H dependent only on the momenta, the relationship between T,
P, and V must be P = (kN/V)T = nRT/V, which is the equation of state
of a perfect gas. Postulate (171) does indeed have ties with "reality "
The
Microcanonical
Ensemble
We postponed discussing exactly what volume of momentum space
was represented by the quantity ft p of Eq. (178), because we did not
need to discuss it at that point. Before we can evaluate ftp we must
take into account the requirements of Liouville's theorem, men
tioned in the paragraphs following Eq. (163). There it was shown
that if the distribution function f(q,p) had the same value for every
point in phase space at which the energy of the system is the same
(i.e., over a surface of constant energy), then f will be independent
of time. Thus the finite volume of phase space occupied by the en
semble of Eq. (176) must be a region between two constant energy
surfaces, or else just the "area" of a single constantenergy sur
face.
Example of a Simple Crystal
The latter alternative, an ensemble of systems, all of which have
the same energy, is one that should be investigated further. Since the
"volume" of phase space occupied by such an ensemble is finite, the
discussion of the previous section applies and for maximum entropy
the distribution function should be constant over the entire constant
energy surface. The resulting ensemble, for which f(q,p) = unless
H(q,p) = U (or f v = unless E^ = U, for the case where quantum
theory is used), Eq. (176) becomes
f = 1/ft (or 1/W) when H(q,p) = U (or when E^ = U)
zero otherwise and S = k ln(ft/h</>) (or k InW) (181)
and is called the microcanonical ensemble. Quantity ft is the "area"
of the surface in phase space for which H(q,p) = U and W is the
total number of quantum states that have allowed energies E v = U
(and, when ft is large enough, ft  h$W).
154
THE MICROCANONICAL ENSEMBLE 155
We shall work out the microcanonical distribution for two simple
systems. The first is the simple crystal model of Eq. (1312), with
each of the N atoms in the crystal lattice being a threedimensional
harmonic oscillator with frequency co/277. Here we shall use quan
tum theory, since the formula for the allowed energies of a quantized
oscillator is simple. The allowed energy for the ith degree of free
dom is h(jo(vi + 1/2), where h = h/27r and i/j is the quantum number
for the ith degree of freedom. Therefore the allowed energy of vi
bration of this crystal is the sum
E p =ha> Y] H +2^ ha;; 0=3N (182)
and the total internal energy, including the potential energy of static
compression [see Eq. (1315)] is
U = Ej, + [(V  V ) 2 /2kV ] =Ho)M +nHo> + [(V  V ) 2 /2kV ]
(183)
where M = (v 1 + v 2 + ••• + v^) and = 3N.
A microcanonical ensemble would consist of equal proportions of
all the states for which M is a constant integer, and W is the num
ber of different permutations of the quantum numbers v\ whose sum
is M. This number can be obtained by induction. When 0=1, there
is only one state for which v 1 = M, so W = 1. When 0=2, there are
M + 1 different states for which v 1 + v 2 = M, one for v 1 =0, v 2 = M,
another for v L = 1, v 2 = M  1, and so on to v 1 = M, v 2 = 0. When
0=3, there are M + 1 different combinations of v 2 and v z when
v x is 0, M different ones when v x = 1, and so on, so that
W = (M + 1) + M + (M  1) + — + 2 + 1 =(M + 1)(M + 2)
for 0=3
Continuing as before, we soon see that for different i^s (i.e.,
degrees of freedom),
(M+0  1)! _ (M + 3N 1)! rift 4 x
W M!(0  1)! " M!(3N 1)! U5 "* ;
From this and from Eq. (183) we can obtain W as a function of U;
156
STATISTICAL MECHANICS
then from Eq. (181) we can obtain S as a function of U. Probability
i v that the system has any one of the combinations of the v's which
add up to M is 1/W.
Since both M and W are large integers, we can use the asymptotic
formula for the factorial function,
(27in) 1/2 n n e n , n»l
(185)
which is called Stirling's formula. Using it, we can obtain a simple
approximation for the number W of different quantum states which
have the same value of M, and thus of U:
W
M + d>  1
27rM(2>
6
(M + d> 1) M + Ci)  1 M M (0  l)0 +1
2ttM(M + <b\
v*r(»^
(186)
where, since & = 3N is large, we have substituted for 01.
Therefore the entropy of the simple crystal is
S = klnW^kMln(l +^l 
*i.(i + M)
(187)
where we have neglected the logarithm of the square root, since it
is so much smaller than the other two terms.
Let us first consider the highenergy case, where the average
oscillator is in a higher excited state (i/j ^> 1) and therefore where
M ^> $ = 3N. In this case, we can neglect 1 compared to M/<p and
we can use the approximation ln(l + x)  x, good for x <C 1, obtaining
S « k6 + k& ln(M/d>) = k<b ln(eM/#)
so that
(188)
U  3^e S / 3kN +Nhw + [(V  V ) 2 /2kV ], = 3N «
M
Remembering that T = (aU/8S) v , we can find T as a function of S
and then U as a function of T and V, for this hightemperature case.
T =fiw(aM/aS) v M (ho)/ek)e s / 3kN
so that
(189)
U  3NkT +lSma) + [(V  V ) 2 /2kV o ;
THE MICROCANONICAL ENSEMBLE 157
This is to be compared with Eq. (1315). There is a "zeropoint' '
energy (3/2)Nhu> additional to this formula, but it is small compared
to the first term at high temperatures (kT ^> hoj).
At very low energies we have >> M, in which case we can write
S^kM
i / 3eN \
ln Ud
so that
1 = 1 (dS\ ^_k_
(3
w "
or
Jid)M
_. i im , tut 1 / 3eN> \ * k 1 (™\
+ kM = kMln(— J or =m[— J
or
or
M = 3Ne Rw / kT
U  3Nhw /eWkT + *\ + [ (v _ v o ) 2 /2/<V ]
(1810)
Thus as T — (M — 0) the entropy and the heat capacity (dU/3T) v
both go to zero, a major point of difference from the results of Chap
ter 13. We shall see later [the discussion of Eq. (1911), for ex
ample] that this vanishing of S and of C v as T — is a character
istic of systems when classical physics is no longer a good approxi
mation, and the effects of quantization are apparent.
Microcanonical Ensemble for a Perfect Gas
In the case of a perfect gas of N point particles in a volume V of
"normal" size, the energy levels are so closely spaced that we can
use classical physics for temperatures greater than a fraction of a
degree Kelvin. Thus a microcanonical ensemble for such a system is
represented by a distribution function f(q,p) which is zero every
where in phase space except on the "surf ace,' '
158 STATISTICAL MECHANICS
3N
H = £ (l/2m)p?  U (a constant) (1811)
i = l
where it is 1/ft, ft being the integral of dVq dVp over this surface
(i.e., ft is the "area" of the surface). The entropy of the micro
canonical ensemble is then given by Eq. (181). This classical ap
proximation should be valid for T ^> 0.01 °K, as will be shown later
[see the discussion following Eq. (215)].
Since the energy of a perfect gas is independent of the positions of
the particles, the integral of dVq, as shown in Eq. (177), is simply
the nth power of the volume V of the container. The integral of dV p ,
however, is the "area" of the surface in momentum space defined by
Eq. (1811). This surface is the 3Ndimensional generalization of a
spherical surface; the coordinates are the p's and the radius is
R = (2mU) V2 ,
Pl 2 +p + ... + p 2 ) = R 2 ; 0=3N; R 2 = 2mU
Once the area ftp of this hyper spherical surface is computed, the rest
of the calculation is easy, for the volume of phase space occupied is
ft = V N ft p and f and S are given by Eq. (1711).
To find the area we need to define some hyperspherical coordi
nates. Working by induction:
For two dimensions:
Xi = R cos 1? x 2 = R sin 0i, x? + x = R 2
Element of length of circle: ds = R d0i
For three dimensions:
Xi = R cos0i, x 2 = R sin 0i cos0 2 , x 3 = R sin0i sin0 2
Element of area of sphere: dA = R 2 sin0i d0 x d0 2
For four dimensions:
Xi = R cos 0i, x 2 = R sin 0i cos0 2 , x 3 = R sin0i sin0 2 cos0 3 ,
x 4 = R sin 0i sin0 2 sin0 3
Element of surface: dA = R 3 sin 2 0! sin0 2 d0 x d0 2 d0 3
THE MICROCANONICAL ENSEMBLE 159
For $ dimensions:
x x = R cos0i, x 2 = R sin0 x cos0 2 , ... (1812)
x^ _ i = R sin0i"Sin00 _ 2 cos00 _ i, x^ = R sin0 x ••• sin00 _ \
dA = R^ " 1 sin^ ~ 2 0i sin0 ~ 3 2 ••• sin00 _ 2 d0i d0 2 — d0^ _ i
where angle 001 goes from to In and angles 0i,...,002 go
from to 77. To integrate this area element we need the formula
/sin n 9 de = VF[(fn  )j/(n) l] (1813)
where m ! is the factorial function
oo
ml =/x m e x dx = m • (m  1)!; 0! = 1! =1
(i)l=VF=2g)l (1814)
with asymptotic values given by Stirling's formula (185). Thus the
t otal area of the hypersphere is [neglecting such factors as V2 and
VI (1/0)]
77 77 277
A = ftp = R0  1JW " 2 B x d6 x  J sin0 . 2 d V 2 / d ^  1
fi* ')'&* 4)' 6)>
_ 2u(4>/ 2 >R<Z>  1
(}*l)l
„, pmU \(» " X )/2 / _ 2_x(*/2) e(0 _ 2)/2
^WJej(*/2) ; ^ (lg _ 15)
160 STATISTICAL MECHANICS
and the final expression for the volume of phase space occupied is
fl  V N (47rmUe/3N)( 3 / 2 ) N (1816)
where we have used Eq. (185), have replaced [(<£ l)/2] by
(1/2)0 = (3/2)N and have used the limiting formula for the exponen
tial function
(l +~) — e x ; n*°° (1817)
and the e in the formula for Q is, of course, the base of the natural
logarithm, e = 2.71828.
Consequently the entropy of the gas is, from Eq. (176),
S = Nk ln[v(477mUe/3Nh 2 ) 3/2 ]
or
U = (3Nh a /4irme)V"^ e 2s / 3Nk (1818)
which is to be compared with Eq. (179) as well as with the discus
sion following Eq. (66). As with the discussion of Eq. (179), we can
now obtain the thermodynamic temperature and pressure,
T = (3U/3S) V = (h'^irmekJV^e 28 / 3 ^
(1819)
P = _(au/aV) s = (Nh 2 /27rme)V^ 3 e 2s / 3Nk = NkT/V
Also, U =  NkT C v = (9 U/8 T) v = f Nk
So the microcanonical ensemble does reproduce the thermodynamic
behavior of a perfect gas, in complete detail. With Eq. (179) we
were able to obtain the equation of state, but now that we have com
puted the dependence of Qp on U and N, the theoretical model also
correctly predicts the dependence of U on T and thence the heat
capacity of the gas.
The Maxwell Distribution
The microcanonical ensemble can also predict the velocity dis
tribution of molecules in the gas; if we have the distribution function
for the whole gas we can obtain from it the distribution function for
a constituent particle. Utilizing Eq. (1819) we can restate our results
THE MICROCANONICAL ENSEMBLE 161
thus far. Since R = (2mU) ]/2 and U = (3/2)NkT, we can say that the
systems in the microcanonical ensemble for a perfect gas of point
particles are uniformly distributed on the surface of a hypersphere
in 3 N dimensional momentum space of radius (3NmkT) 1/2 . The prob
ability that the point representing the system is within any given re
gion dA on this surface is equal to the ratio of the area of the region
dA to the total area A = fip of the surface. The region near where
the pi axis cuts the surface, for example, corresponds to the micro
states in which the x component of momentum of particle 1 carries
practically all the energy (l/2)0kT of the whole system. It is an
interesting property of hypersphere s of large dimensionality (as we
shall show) that the areas close to any axis are negligibly small com
pared to the areas well away from any axis (where the energy is
relatively evenly divided between all the degrees of freedom). There
fore the chance that one degree of freedom will turn out to have most
of the energy of the whole gas, (l/2)<£kT, and that the other components
of momentum are zero is negligibly small.
To show this, and incidentally to provide yet another ' 'derivation' '
of the Maxwell distribution, we note that the probability that the
momentum coordinate, which we happen to have labeled by the sub
script 1, has a value between p x and pi + dpi can be obtained easily,
since we have chosen our angle coordinates such that pi = R cos#i.
Thus the probability
dA = 1 (1/2U ! Sin ^ " 2 9l d01 Sin ^ ~ 3 ° 2 d ° 3
... sin 6> _ 2 d6>0 _ 2 dfy _ 1 (1820)
as a function of 9 l9 is only large near Q 1 = (1/2)77 [i.e., where pi =
(0mkT) 2 cos 61 is very small compared to (0mkT) 2 ] and drops off
very rapidly, because of the large power of sin#i, whenever the mag
nitude of pi increases. The factor sin0 %6i ensures that the prob
ability is very small that the degree of freedom labeled 1 carries
most of the total kinetic energy (l/2)0kT of the gas. This would be
true for each degree of freedom. It is much more likely that each
degree of freedom carries an approximately equal share, each having
an amount near (l/2)kT.
The formula for the probability that degree of freedom 1 have
momentum between pi and p x + dpi, irrespective of the values of the
other momenta, is obtained by integrating dA/A over 2 , 3 ,...,0a _ \.
Using the results of Eqs. (1813) to (1817) produces
162 STATISTICAL MECHANICS
1 [(1/2)01]! / _ p 2 \(0~3)/2
(7T0mkT) V2 [(1/2)0  3/2] ! \ 0mkT/ Pl
since d^ = (l/^mkT^Mdpi/sinfli) and sin 2 6 l = 1  (p?/0mkT). To
obtain the Maxwell distribution in its usual form we utilize Eqs.
(185) and (1817) and consider factors like [l  {2/<$>)Y 2 and
[l  (p?/(/)mkT)]" 3/2 to equal unity [but not such factors to the (1/2)0
power, of course]. The calculations go as follows:
f(Pl) dPl * (ZirrnkT)*
1  (2/0)
1  (3/0)
(*/2) . 2
x[l  (p!/0mkT)]^ 3 )/ 2 d Pl
" (27rmkT) V2 ex P(Pi/2mkT) d Pl (1821)
which is the familiar Maxwell distribution for one degree of freedom
[see Eq. (127)].
This time we arrived at the Maxwell distribution as a consequence
of requiring that the total kinetic energy of the gas be (l/2)0kT and
that all possible distributions of this energy between the degrees
of freedom be equally likely. For = 3N large, by far the majority
of these configurations represent the energy being divided more or
less equally between all degrees of freedom, with a variance for
each p equal to 2m times the mean kinetic energy per degree of
freedom, (l/2)kT. We note that the Maxwell distribution is not valid
unless the individual atoms are, most of the time, unaffected by the
other atoms, mutual collisions being rare events.
The Canonical
Ensemble
The microcanonical ensemble has sufficed to demonstrate that the
basic postulates of statistical mechanics correspond to the facts of
thermodynamics as well as of kinetic theory. But it has several draw
backs, hindering its general use. In the first place, the computation
of the number of microstates that have a given energy is not always
easy. It actually would be easier to calculate average values with a
distribution function that included a range of energies, rather than one
that differs from zero only when the energy has a specific value.
In the second place (and perhaps more importantly) the micro
canonical ensemble corresponds to a system with energy U, com
pletely isolated from the rest of the universe, which is not the way a
thermodynamic system is usually prepared. We usually do not know
the exact value of the system's energy; we much more often know its
temperature, which means that we know its average energy. In other
words, we do not usually deal with completely isolated systems, but
we do often deal with systems kept in contact with a heat reservoir
at a given temperature, so that its energy varies somewhat from in
stant to instant, but its time average is known. This changes the
boundary conditions of Eq. (172) and the resulting distribution func
tion will differ from that of Eq. (181).
Solving for the Distribution Function
Suppose we prepare an ensemble as follows: Each system has the
same number of particles N and has the same forces acting on the
particles. Each system is placed in a furnace and brought to equi
librium at a specified temperature, with each system enclosed in a
volume V. Thus, although we do not know the exact energy of any
single system, we do require that the mean energy, averaged over
the ensemble, has the relationships between S and T expressed in
Eqs. (63) and (88), for example. The distribution function for such
163
164 STATISTICAL MECHANICS
an ensemble, corresponding to a system in contact with a heat
reservoir, should satisfy the following requirements:
S = kZ) i v l n fy is maximum,
" v r (19 " 1)
subject to Li v iy 1 and £j p i v E p = \l,
the internal energy.
We solve for i v by using Lagrange multipliers. We require
S + ot Yj v i v + Oi e Tj u fyEp be maximum,
with a and a e adjusted so that (192)
L v iu= 1 and E v i v E p = V t
where U, for example, satisfies the equation U =F + TS. Setting the
partials of this function, with respect to the f's, equal to zero we ob
tain
k lnfj,  k + a + o^e^v =
or
i v = exp[(a  k + a e E p )/k] (193)
The value of the Lagrange multiplier a is adjusted to satisfy the
first subsidiary condition,
e (ao/k)lj^ e a e E v /k= 1; e a cA = e/Z; Z =J^ e a eE u /k
The value of the Lagrange multiplier a e is obtained by requiring that
the sum Yj^v^v should behave like the thermodynamic potential U.
For example, the sum we are calling the entropy is related to the sum
we are calling U, by virtue of Eq. (193), as follows:
S =kSf y Infy = EMao  k +a e E v ) = fao  k)  a e U
v v
where we have used Eq. (193) for lnfj, and also have used the fact
that the f's must satisfy Yj^v = * ( tne firs t subsidiary condition).
However, if U is to be the thermodynamic potential of Eqs. (63)
and (88), this relation between S and U should correspond to the
equation S = (F + U)/T. Therefore the Lagrange multipliers must
have the following values:
a k = F/T; a e = (1/T)
The solution of requirements (191) is therefore
THE CANONICAL ENSEMBLE 165
t v = (l/Z)e E ^ kT ; Z =Ee" E ^ kT = e"F/kT
S=kEf l /[lnZ +(E^/kT)]=2^=(f)
(194)
The ensemble corresponding to this distribution is called the
canonical ensemble. The normalizing constant Z, considered as a
function of T and V, is called the partition function. Part of the
computational advantage of the canonical ensemble is the fact that
all the thermodynamic functions can be computed from the partition
function. For example,
F = kTlnZ; S = (8F/8T) V ; P = (3F/8V) T (195)
When the separations between successive allowed energies E v
are considerably less than kT, classical mechanics can be used and
instead of sums over the quantum states v of the system we can use
integrals over phase space. The distribution function is the proba
bility density f(q,p), and, for a system with </> degrees of freedom,
f(q,p) = (l/h0Z)e H fa>P)/ kT
(196)
Z = (l/h0)/ /eH/kT dVq dV p
where H(q,p) is the Hamiltonian function of the system, the kinetic
plus potential energy, expressed in terms of the q's and p's [see
Eqs. (139) and (161)]. From Z one can then obtain F, S, etc., as
per Eq. (195). The H of Eq. (196) is the total energy of the system,
whereas the H of Eq. (1310) is the energy of a single molecule. One
might say that the canonical distribution function is the Maxwell 
Boltzmann distribution for a whole system. It is an exact solution,
whereas the f of Eq. (1310) for a molecule is only valid in the limit
of vanishing interactions between molecules.
General Properties of the Canonical Ensemble
The first thing to notice about this ensemble is that the distribu
tion function is not constant; all values of the energy are present, but
some of them are less likely to occur than others. In general the
larger the energy H the smaller is f. However, to find out the prob
ability that the system has a particular value of the energy, we must
multiply f by ft(H), the "area" of the surface of constant H in phase
space. This usually increases rapidly with H; for example £2 = Wh^ 
(27reH/3Nco) 3N for the simple crystal of Eq. (188) and ft ^
V N (47rmHe/3N)(3/2)N f or the perfect gas of Eq. (1816). The product
166 STATISTICAL MECHANICS
^f(QjP) K ^(H)e~ H A T at first increases and then, for H large enough,
the exponential function "takes over" and ffi eventually drops to zero
as H°°.
The value of H that has the most representatives in the ensemble
is the value for which £2e~"H/kT is maximum. For the gas this value
is H = (3/2)NkT and for the crystal it is 3NkT; in each case it is
equal to the average value U of energy of the ensemble. The number
of systems in the ensemble with energy larger or smaller than this
mean value U diminishes quite sharply as H  Ul increases. Al
though some systems with H * U do occur, the mean fractional devi
ation from the mean (AH/U) of the canonical distribution turns out to
be inversely proportional to V0 and thus is quite small when $ is
large. Therefore the canonical ensemble really does not differ very
much from the microcanonical ensemble; the chief difference is that
it often is easier to handle mathematically.
The advantages of this greater ease are immediately apparent
when we wish to determine the general properties of a canonical en
semble. For these can be deduced the properties of the partition
function Z. For example, in many cases the system consists of N
subsystems, the ith having 6^ degrees of freedom (so that =
2^i = l 6 i^ each subs Y stem having negligible interaction with any
other, although there may be strong forces holding each subsystem
together. For a perfect gas of N molecules, the molecules are the
subsystems, the number of degrees of freedom of each molecule
being three times the number of particles per molecule. For a
tightly bound crystal lattice the "subsystems" are the different
normal modes of vibration of the crystal— and so on. Whenever such
a separation is possible, the partition function turns out to be a
product of N factors, one for each subsystem.
To see why this is so, we note that if the subsystems are mutually
independent, the Hamiltonian of the system is a sum of N separate
terms, H =S?LiHj, where Hj is the energy of the jth subsystem
and is independent of the coordinates of any other subsystem. For a
quantized system, the energy Ev 1 v 2 '"V(t is a sum of N separate
terms, the term Ej being the allowed energy of the jth subsystem,
dependent on 6j quantum numbers only (call them v],v] +l,...,^j +6p
and independent of all the quantum numbers of the other subsystems.
The partition function is the sum of exp(Ezy 1 ^ 2 ...y ( f ) /kT) =
e E 1 /kT... e Ej/kT... e EN/kT over ^11 quantum states of all sub
systems,
Z= Jj exp(Ei/ 1 j/ a ...i/ ( j ) /kT) =Ziz 2 —zj...zn
all v
where (197)
zj =X>xp[Ej(i/j,i/j +1 ,...,j,j + 6j)/kT]
THE CANONICAL ENSEMBLE 167
the sum for zj being over all the quantum numbers of the jth sub
system.
For example, the energy of interaction between the magnetic field
and the orientation of the atomic magnets in a paramagnetic solid is,
to a good approximation, independent of the motion of translation or
vibration of these and other atoms in the crystal. Consequently the
magnetic term in the Hamiltonian, the corresponding factor in the
partition function, and the resulting additive terms in F and S can
be discussed and calculated separately from all the other factors and
terms required to describe the thermodynamic properties of the
paramagnetic material. This of course is what was done in Chapter
13 [see Eqs. (1316) to (1318)].
The Effects of Quantization
To follow this general discussion further, we need to say some
thing about the distribution of the quantized energy levels of the jth
subsystem. There will always be a lowest allowed level, which we
can call E} 9 1. This may be multiple, of course; there may be gji
different quantum states, all with this same lowest energy. The next
lowest energy can be labeled Ej,2; it may have multiplicity gj2; and
so on. Thus we have replaced the set of 6j quantum numbers for the
jth subsystem by the single index number v, which runs from 1 to
°°, and for which Ej^ + i > Ej^, the vth level having multiplicity gj^.
Thus the jth factor in the partition function can be written
zj = S gjye E JiV kT (198)
v = l
If the energy differences Ej2  Eji and Ej3  Ej2, between the
lowest three allowed energy levels of the jth subsystem are quite
large compared to kT, then the second term in the sum for zj is
small compared to the first and the third term is appreciably
smaller yet, so that
zj ~gjieEjl/kT[i +(g j2 /gjl)e( E j2Ejl)/kT] ( 19 . 9 )
for kT small compared to Ej2  Eji. The factor in brackets be
comes practically independent of T when kT is small enough.
The Helmholtz function for the system is a sum of terms, one
for each subsystem,
N
F = kT InZ = 2 Fj; F^kTlnzi (1910)
j = l
168 STATISTICAL MECHANICS
and the entropy, pressure, and the other thermodynamic potentials
are then also sums of terms, one for each subsystem. Whenever any
one of the subsystems has energy levels separated farther apart than
kT. the corresponding terms in F, S, and U have the limiting forms,
obtained from Eq. (199),
Fj  kT lngji +Ejl  (gj2/gjl)kTe( E j2Ejl)/kT
Sj k lngji +(gj2/gjl
k +
Ej2  gjl
r (E j2 Eji)/kT (19 _ n)
Uj
F j + SjT
Ejl +(gj2/gjl)(Ej 2  Ejl)e(Ej2Ejl)/kT
Thus whenever the jth subsystem has a single lowest state (gji =
1, i.e., when the subsystem is a simple one) its entropy goes to zero
when T is reduced so that kT is much smaller than the energy sep
aration between the two lowest quantum states of the subsystem. On
the other hand, if the lowest state is multiple, S goes to k In gjl as
T — 0. In either case, however, the heat capacity Cjv = (cUj/cT) v of
the subsystem vanishes at T = 0. Since all the subsystems have non
zero separations between their energy levels, these results apply to
all the subsystems, and thus to the whole system, when T is made
small enough. We have thus "explained" the shape of the curve of
Fig. 31 and the statements made at the beginning of Chapter 9 and
the discussion following Eq. (1810).
The High Temperature Limit
When T is large enough so that many allowed levels of a sub
system are contained in a range of energy equal to kT, the exponen
tials in the partition function sum of Eq. (198) vary slowly enough
with v so that the sum can be changed to a classical integral over
phase space, of the form given in Eq. (196). In this case, of course,
the dependence of Z on T and V is determined by the dependence of
the Hamiltonian H on p and q. For example, if the subsystem is a
particle in a perfect gas occupying a volume V, Hj = (pj x + Pjy +
pj z )> 2m depends only on the momentum, and the factor in the parti
tion function for the jth particle is
zj = (1, h 3 ) Jj"/dV q /j/exp [(p^ + p + p)/2mkT] dV p
= (V/h 3 )(2mkT) 32 (1912)
and if there are N particles,
NkT InV fNkT ln&mkT/h 2 ); U
[but see Eq. (2113)]
NkT
THE CANONICAL ENSEMBLE 169
On the other hand, if the ''subsystem" is one of the normal modes
of vibration of a crystal, Hj = (pj/2m) + (mwjqj/2) depends on both q
and p, so that
Zj = (l/h)/e m ^J ?c l!/ 2kT dqj / e Pj/ 2mkT d PJ
= 277kT/hct>j = kT/hu>j (1913)
and, if there are 3N modes
3N
F = kT £ ln(hwi)  3NkT In (kT); U = 3NkT
j=l
the difference between U = (3/2)NkT and U = 3NkT being caused by
the presence of the q's in the expression for H in the latter case.
For intermediate temperatures we may have to use equations like
(1911) for those subsystems with widely spaced levels and classical
equations like (1912) or (1913) for those with closely packed energy
levels. The mean energy of the former subsystems is practically in
dependent of T, whereas the mean energy of the latter depends lin
early on T: thus only the latter contribute appreciably to the heat
capacity of the whole. In a gas of diatomic molecules, for example,
the energy levels of translational motion of the molecules are very
closely packed, so that for T larger than 1°K, the classical integrals
are valid for the translational motions, but the rotational, vibrational,
and electronic motions only contribute to C v at higher temperatures.
20
Statistical
Mechanics
of a Crystal
Two examples of the use of the canonical ensemble will be dis
cussed here; the thermal properties of a crystal lattice and those
of a diatomic gas. Both of these systems have been discussed before,
but we now have developed the techniques to enable us to work out
their properties in detail and to answer the various questions and
paradoxes that have been raised earlier.
Normal Modes of Crystal Vibration
For example, the simplified crystal model of Eq. (1315) assumed
that each atom in a crystal vibrated independently of any other and
thus that every atom had the same frequency of vibration. This is
obviously a poor approximation for a real crystal, and in this chap
ter we shall investigate a model in which the effects of one atom's
motion on its near neighbors are included, at least approximately.
We shall find that this slightly improved model, although still quite
simplified, corresponds surprisingly well to the measured thermal
behavior of most crystals.
For comparison, however, we shall complete our discussion of
the crystal model of Eq. (1315), with no interaction between atoms
and with all atomic frequencies equal. The allowed energy for the
jth degree of freedom is hu(yj + 1/2) (where v^ is an integer) and
the allowed energy of the whole system is
S v\ + E ; E = ±<f)fiu) + [(V  V ) 2 /2kV ]
3=1 2
E^,...,^ =fiw
and therefore the partition function, Helmholtz function, and other
quantities are
170
STATISTICAL MECHANICS OF A CRYSTAL 171
z = e Eo/kT ZiZ2 ... Z0 . zj= ge^^jATMleWkT)
F = kT In Z = E + 3NkT In (1  e W kT )
S = 3Nk ln(l  e W kT ) + [(3Nnw/T)/(e n ^/ kT  1)]
(201)
U = E + [3NiW(e*W kT . i)]
{
E + 3Nhwe WkT kT « nw
E + 3NkT kT > na>
C\
= 3N(na?) 2 eW k T _ J(3NhV/kT 2 )e WkT kT <<c ftw
v kT 2( e ti w /kT _ i) 2 1 3Nk kT > nw
Where we have used the formula
£x n = l/(lx), x<l (202)
n =
to reduce the partition sums to closed formulas. These equations
were first obtained by Einstein. The heat capacity C v is plotted in
Fig. 201 as the dashed curve. It does go to zero as T becomes
much smaller than "ftco/k, as we showed in Chapter 19 that all quan
tized systems must. Actually it goes to zero more decidedly than the
experimental results show actual crystals do. We shall soon see that
this discrepancy is caused by our model's neglect of atomic interac
tions.
In an actual crystal the interaction forces between the atoms, which
tend to bring each atom back to its equilibrium position, depend in a
complicated way on the displacements of whole groups of atoms. If
the displacements are small, the forces depend linearly on the rela
tive displacements and thus the potential energy is a combination of
quadratic terms like (l/2)K^qf, depending on the displacements from
equilibrium of one of the atoms [which were included in the simplified
model of Eq. (201)] but also terms like (l/2)Kij(qi  qi) 2 , corre
sponding to a force of interaction between one atom and another. Al
though many of the Kjj's are small or zero, some are not. The total
potential energy is thus
172
STATISTICAL MECHANICS
_ T
..,,..._
0.5
—
Debye ^ ff
—
.*
Jl
—
fc
7/
CO
\
J . "^Einstein
>
—
O
//
//
//
//
0.1

//
/ /
/ /
/ /
/ /
_
0.05



/ /
7 7

—
A /
/ /
—
_ ,
r f
/
/
_
/
' 1 II
0.01
/ l
l/l 1 1 1 1 1 1 i 1 1 1 1
1 1 1 1 1 1 1 1
0.05
0.1
0.5
Fig. 201. Specificheat curves for a crystal. Ordinate
y for the Debye curve is T/0 = kT/hu> m ; or
dinate for the Einstein curve is 3kT/4hco.
Circles are experimental points for graphite,
triangles for KC1.
3N
IE
i = l
3N
KiQi + J] K ij (( *i " *$
j>i
3N
ij=l
An = Ki + 2 £ K ij; Aij = Aji = Kij
Therefore the Hamiltonian for the crystal is
STATISTICAL MECHANICS OF A CRYSTAL 173
3N 3N
H = ^L^ + \L A Wm (203)
3=1 i,J=l
Actually there are six coordinates not represented in the sum over
the q's, those for the motion of the crystal as a rigid body; so the
total number of coordinates in the second sum is 3N6 rather than
3N. However, 6 is so much smaller than 3N that we can ignore this
discrepancy between the sums, by leaving out the kinetic energy of
rigid motion and calling 3N6 the same as 3N.
The solution of a dynamical problem of this sort is discussed in
all texts of dynamics. The matrix of coefficients Aij determines a
set of normal coordinates, Q n , with conjugate momenta P n , in terms
of which the Hamiltonian becomes a sum of separated terms, each of
which is dependent on just one coordinate pair,
3N
H =2] [d/ m n)Pn+ mnconQn] + [(V  V ) 2 /2kV ] (204)
n = l
Application of Hamilton's equations (161), (9H/3P n ) = Q n and
(3H/8Q n ) = £ n , results in a set of equations
P n = m n Q n ; Qn + w n Qn = (205)
which may be solved to obtain the classical solution Q n = Q0ne iwnt .
Thus u n /27r is the frequency of oscillation of the nth normal mode
of oscillation of the crystal.
These normal modes of the crystal are its various standing waves
of free vibration. The lowest frequencies are in the sonic range, cor
responding to wavelengths a half or a third or a tenth of the dimensions
of the crystal. The highest frequencies are in the infrared and corre
spond to wavelengths of the size of the interatomic distances. Be
cause there are 3N degrees of freedom there are 3N different stand
ing waves (or rather 3N6 of them, to be pedantically accurate); some
of them are compressional waves and some are shear waves.
Quantum States for the Normal Modes
According to Eq. (167), the allowed energies of a single normal
mode, with Hamiltonian (l/2mj)Pj + (l/2)mjcjjQj are given by the for
mula hwJi'j + (1/2)], where v^ is an integer, the quantum number of
the jth normal mode. Sometimes the quantized standing waves are
174 STATISTICAL MECHANICS
called phonons; v^ is the number of phonons in the jth wave. Micro
state v of the crystal corresponds to a particular choice of value for
each of the iVs. The energy of the phonons in microstate v is then
3N 3N
E„ = E (V) +h £ ">)vy, Eo = [(V  Vo)/2KV ]+h^ o>j
j=1 ' Jl
(206)
each term in the sum being the energy of a different standing wave.
The difference between this and the less accurate Einstein formulas
of Eqs. (201) is that in the previous case the co's were the same
for all the oscillators, whereas inclusion of atomic interaction in the
present model has spread out the resonant frequencies, so that each
standing wave has a different value of u>.
According to Eq. (194) the partition function is
/JCo ll Z.COjZAjN
2 exp 1 j= e~ E o/ kT z 1 z 2 z3N
all i/j's
where
Zi =Se fiw J^/ kT = (1  e n ^j/kT)i (207)
"J
and thus, from Eq. (195), the Helmholtz function for the crystal is
3N
F = kT InZ = E (V) +kT S ln(l  e~ ft wj/kT) (208)
1=1
We can then compute the probability i v that the system is in the
microstate specified by the quantum numbers u = v lf v 2 > ••• ^3N Jt
is the product [see Eq. (194)].
f I , = (l/Z)eE l ,/kT =fl f 2f3 ... f 3 N
where
fj = (l/zjJe^J^/H = e^j^jAT  e"^j(^j +l)/kT (209)
is the probability that the jth standing wave of thermal vibration is
STATISTICAL MECHANICS OF A CRYSTAL 175
in the v$th quantum state. The probability that the crystal is in the
microstate v is of course the product of the probabilities that the
various normal modes are in their corresponding states.
When kT is small compared to hwj for all the standing waves of
crystal vibration, all the zj's are practically unity, F is approxi
mately equal to E (V), independent of T, and the entropy is very
small. When kT is large compared to any ho)j, each of the terms in
parentheses in Eq. (208) will be approximately equal to hu)j/kT and
consequently the Helmholtz function will contain a term 3NkT ln(kT),
the temperature dependent term in the entropy will be 3Nk InkT, and
the heat capacity will be 3Nk = 3nR, as expected. To find values for
the intermediate temperatures we must carry out the summation over
j in Eq. (208) or, what is satisfactory here, we must approximate
the summation by an integral and then carry out the integration.
Summing over the Normal Modes
The crucial question in changing from sum to integral is: How
many standing waves are there with frequencies (times 2n) between
(jo and & + dw? There are three kinds of waves in a crystal, a set of
compressional waves and two sets of mutually perpendicular shear
waves. If the crystal is a rectangular parallelopiped of dimensions
lx, ly, lz, the pressure distribution of one of the compressional waves
would be
p = ojQj sin(7rkjx/l x ) sin(7rmjy/ly) sin(7rnjz/l z )
where Qj(t) is the amplitude of the normal mode j, with equations of
motion (205), a is the proportionality constant relating Qj and the
pressure amplitude of the compressional wave, and kj,mj,nj are in
tegers giving the number of standingwave nodes along the x, y, and z
axes, respectively, for the jth wave.
The value of cdj, 2ti times the frequency of the jth mode, is given
by the familiar formula
w] = (77ckj/l x ) 2 + (7rcmj/l y ) 2 + (7rcnj/l z ) 2 (2010)
where c is the velocity of the compressional wave. Each different j
corresponds to a different trio of integers kj,mj,nj. A similar dis
cussion will arrive at a similar formula for each of the shearwave
sets, except that the value of c is that appropriate for shear waves.
The problem is to determine how many allowed u>\'s have values be
tween u) and go +du.
To visualize the problem, imagine the allowed cuj's to be plotted
as points in "w space," as shown in Fig. 202. They form a lattice
176
STATISTICAL MECHANICS
Representation of allowed values of co in cu
space.
of points in the first octant of the space, with a spacing in the "u) x "
direction of 7rc/l x , a spacing in the "u>y" direction of 7rc/ly, and a
spacing in the "w z " direction of ttc/1 z , with the allowed value of co
given by the distance from the origin to the point in question, as
shown by the form of Eq. (2010). The point closest to the origin can
be labeled j = 1, the next j = 2, etc. The spacing between the allowed
points is therefore such that there are, on the average, l x l y l z /7r 3 c 3 =
V/u 3 c 3 points in a unit volume of "oj space," where V = lxlylz * s th e
volume occupied by the crystal.
Therefore all the allowed coj's having value less than co are rep
resented by those points inside a sphere of radius co (with center at
the origin). The volume of the part of the sphere in the first octant is
(1/8)(47tw 3 /3) and, because there are VA 3 c 3 allowed points per unit
volume, there must be (V/77 3 c 3 )(7ru> 3 /6) standing waves with values of
ooi less than u>. Differentiating this with respect to a>, we see that the
average number of goj's with value between co and u> + du> is
dj = (V/2t7 2 c 3 )co 2 dw
(2011)
Several comments must be made about this formula. In the first
place, the formula is for just one of the three sets of standing waves,
and thus the dj for all the normal modes is the sum of three such
formulas, each with its appropriate value of c, the wave velocity.
But we can combine the three by using an average value of c, and say
that, approximately, the total number of standing waves with a>j's
between w and oo + dco is
STATISTICAL MECHANICS OF A CRYSTAL 177
dj = (3V/27r 2 c 3 )w 2 du> (2012)
where c is an appropriate average of the wave velocities of the com
pressional and shear waves. Next we should note that Eq. (2011)
was derived for a crystal of rectangular shape. However, a more
detailed analysis of standing waves in crystals of moregeneral
shapes shows that these equations still hold for the other shapes as
long as V is the crystal volume. For a differently shaped crystal,
the lattice of allowed points in go space is not that shown in Fig. 202,
but in spite of this the density of allowed points in co space is the
same, V/7T 3 c 3 .
Next we remind ourselves that there is an upper limit to the al
lowed values of the coj's; in fact there can only be 3N different nor
mal modes in a crystal with N atoms (3N6, to be pedantically
exact). Therefore our integrations should go to an upper limit co m ,
where
3N o; m ct> m
3N = E 1 = / dj = (3V/2ti 2 c 3 )/ co 2 du = (Vu> 3 m /27T 2 c 3 )
j=l °
or
w m = (677 2 Nc 3 /V) j/3 (2013)
Finally we note that both Eqs. (2012) and (2013) are approxima
tions of the true state of things, first because we have tacitly as
sumed that c is independent of co, which is not exactly true at the
higher frequencies, and second because we have assumed that the
highest compressional frequency is the same as the highest shear
frequency, namely, co m /27r, and this is not correct either. All we can
do is to hope our approximations tend to average out and that our
final result will correspond reasonably well to the measured facts.
The Debye Formulas
Returning to Eq. (208), we change from a sum over j to an inte
gral over dj, using Eq. (2012) and integrating by parts; we obtain
F = [(V  Vo) 2 /2/cV ] + / Igliwj + kT ln(l  e  nw j/ kT ) dj
= E (V) + (3kTV/27i 2 c 3 ) / ln(l  e'WkT)^ dw
178 STATISTICAL MECHANICS
277 2 c 3 6tt 2 c 3 \ kT
(2014)
where
E 
[(VV )V2kV ]=
dj = (3V?ia>£
,/M"V)
The function D, defined by the
integral
D(x)
X
= (3/x 3 ) / [z 3 dz/(e z
1)]
(V/5X 3
\l  (3/8)x
x» 1
x <T 1
(2015)
is called the De bye function, after the originator of the formula.
We now can express the temperature scale in terms of the Debye
temperature 6 = fiw m /k (which is a function of V) and then write
down the thermodynamic functions of interest,
F = [(VV ) 2 /2kV ] +Nk6 + NkT[3 In (1  e" / T )  D(0/T)]
[(V  V ) 2 /2kV ] +  NkS  (7T 4 NkT 4 /56 3 ) T < 6
[(VV ) 2 /2/cV ] + ±NkO + 3NkT ln(0/T)NkT
2
T»0
S = Nk 3 ln(le^/ T ) + 4D ~
*(47r 4 NkTV50 3 ) T«e
3Nk In (Te 4/3 /£) T>£
U = [(V  V ) 2 /2kV ] + Nk6 + U V (T); U v = 3NkT D(6/T)
(2016)
C v = 3Nk
I . (30/T)
T e */ T l
(^T^NkT 3 ^ 3 ) T<C#
3Nk T >
STATISTICAL MECHANICS OF A CRYSTAL
179
P = [(V  V)/kV ]   Nk0'  3NkT(0'/0)D(0/T)
[(V  V)//cV ]  Nk0'  7r 4 Nk^'(T/e) 4 T «6
[(V V)AV ]  3NkT(0'/0)
T»0
where d' = d0/dV == (h/k)(du> m /dV) is a negative quantity. Referring
to Eq. (36) we see that the empirical equation of state is approxi
mately the same as the last line of Eqs. (2016) if (3Nk0'/0) is equal
to P/k of the empirical formula. This relationship can be used to
predict values of if 0' can be computed, or it can be used to deter
mine 6' from measurements of j3 and k.
The functions D(x) = [xU v (0/x)/3Nk0] and C v /3Nk are given in
Table 201 as functions of x = 0/T.
Table
201
X
D(x)
C v /3Nk
X
D(x)
C v /3Nk
0.0
1.0000
1.0000
4.0
0.1817
0.5031
0.1
0.9627
0.9995
5.0
0.1177
0.3689
0.2
0.9270
0.9980
6.0
0.0776
0.2657
0.5
0.8250
0.9882
8.0
0.0369
0.1382
1.0
0.6745
0.9518
10
0.0193
0.0759
1.5
0.5473
0.8960
12
0.0113
0.0448
2.0
0.4411
0.8259
15
0.0056
0.0230
2.5
0.3540
0.7466
20
0.0024
0.0098
3.0
0.2833
0.6630
25
0.0012
0.0050
A curve of C v /3Nk versus T/0 is given in Fig. 201 (solid curve),
Comparison with Experiment
Several checks with experiment are possible. By adjusting the value
of we can fit the curve for C v , predicted by Eq. (2016) and drawn
in Fig. 201, to the experimental curve. That the fit is excellent can
be seen from the check between the circles and triangles and the solid
line. We see, for example, that the Debye formula, which takes into
account (approximately) the coupling between atomic vibrations, fits
better than the Einstein formula, which neglects interaction, the dis
crepancy being greatest at low temperatures.
From the fit one of course obtains an empirical value of =liu> m /k
for each crystal measured, and thus a value of u) m for each crystal.
However, by actually measuring the standingwave frequencies of the
180
STATISTICAL MECHANICS
crystal and by summing as per Eq. (2013), we can find out what a> m
(and thus 6) ought to be, and then check it against the 6 that gives the
best fit for C v . These checks are also quite good, as can be seen
from Table 202.
Table 202
Substance
0, °K
K,
from C v fitting from elastic data
NaCl
KC1
Ag
Zn
308
230
237
308
320
246
216
305
Thus formulas (2016) represent a very good check with experi
ment for many crystals. A few differences do occur, however, some
of which can be explained by using a somewhat more complicated
model. In a few cases, lithium for example, the normal modes are so
distributed that the approximation of Eq. (2012) for the number of
normal modes with ujj's between w and oj + da> is not good enough,
and a better approximation must be used [which modifies Eqs. (2013)
and (2014)] . In the case of most metals the C v does not fit the Debye
curve at very low temperatures (below about 2°K); in this region the
C v for metals turns out to be more nearly linearly dependent on T
than proportional to T 3 , as the Debye formula predicts. The discrep
ancy is caused by the free electrons present in metals, as will be
shown later.
Statistical
Mechanics
of a Gas
We turn now to the lowdensity gas phase. A gas, filling volume V, is
composed of N similar molecules which are far enough apart so the
forces between molecules are small compared to the forces within a
molecule. At first we assume that the intermolecular forces are neg
ligible. This does not mean that the forces are completely nonexistent;
there must be occasional collisions between molecules so that the gas
can come to equilibrium. We do assume, however, that the collisions
are rare enough so that the mean potential energy of interaction be
tween molecules is negligible compared to the mean kinetic energy of
the molecules.
Factoring the Partition Function
The total energy of the system will therefore be just the sum of the
separate energies e(^ mo i e ) of the individual molecules, each one de
pending only on their own quantum numbers (which we can symbolize
by ^mole) an( * the partition function can be split into N molecular
factors, as explained in Eq.(197):
z= ( z mole) N ; z mole = La exp [€(^ mole )/kT]
^mole
[but see Eq. (2112)].
In this case the partition function can be still further factored, for
the energy of each molecule can be split into an energy of translation
H tr of the molecule as a whole, an energy of rotation H ro j, as a rigid
body, an energy of vibration H yib of the constituent atoms with respect
to the molecular center of mass, and finally an energy of electronic
motion E e ^:
H mole = H tr + H rot + H vib + H el (211)
181
182 STATISTICAL MECHANICS
To the first approximation these energy terms are independent; the
coordinates that describe Ht r , for example, do not enter the functions
H ro t, H v ib, or H e i unless we include the effect of collisions, and we
have assumed this effect to be negligible. This independence is not
strictly true for the effects of rotation, of course; the rotation does
affect the molecular vibration and its electronic states to some extent.
But the effects are usually small and can be neglected to begin with.
Consequently, each molecular partition function can be, approxi
mately, split into four separate factors',
z mole = z tr ' z rot* z vib ' z el
and the partition function for the system can be divided correspond
ingly,
Z= Zt r Z ro t Z v ib Z e i
where
N  N
Ztr = (ztr) > Z rot = (z rot ) , etc. (212)
The individual molecular factors are sums of exponential terms, each
corresponding to a possible state of the individual molecule, with
quantized energies,
z tr = 2j ex P( e kmn/ kT) ; z rot = zL g\v exp(c^VkT)
k,m,n A,f
(213)
and so on, where k,m,n are the quantum numbers for the state of
translational motion of the molecule, \,v those for rotation, etc., and
where g^ are the multiplicities of the rotational states (the g's for
the translational states are all 1, so they are not written).
The energy separation between successive translational states is
very much smaller than the separation between successive rotational
states, and these are usually much smaller than the separations be
tween successive vibrational states of the molecule; the separations
between electronic states are still another order of magnitude larger.
To standardize the formulas, we shall choose the energy origin so the
c for the first state is zero; thus the first term in each z sum is
unity.
The Translational Factor
Therefore there is a range of temperature within which several
terms in the sum for zj r are nonnegligible, but only the first term is
nonnegligible for z rot , z v ^, and z e j. In this range of temperature
STATISTICAL MECHANICS OF A GAS 183
the total partition function for the gas system has the simple form
z tr ~
2 i ex P fCtannATJ
k,m,n
(214)
all the other factors being practically equal to unity. To compute Z
for this range of temperature we first compute the energies ekmn anc *
then carry out the summation. From it we can calculate F, S, etc.,
for a gas of low density at low temperatures .
The Schrbdinger equation (166) for the translational motion of a
molecule of mass M is
ft 2 /2M)[ 2 */ax 2 ) + 2 */ay 2 ) + OVaz 2 )] = * tr *
If the gas is in a rectangular box of dimensions l x ^y^z and volume
V = tyjiylzi with perfectly reflecting walls, the allowed wave functions
and energies turn out to be
^kmn = A sin (irkx/?. x ) • sin (7rmy/£ y ) • sin (7rnz/£ z )
4mn = (^ 2 ^ 2 /2M)[(k/£ x ) 2 + (m/* y ) 2 + (n/* z ) 2 ] = p 2 /2M (215)
where p is the momentum of the molecule in state k,m,n. For a mol
ecule of molecular weight 30 and for a box 1 cm on a side, 7r 2 "h 2 /2M£ 2
« 10" 38 joule. Since k =* 10~ 23 joule/°K, the spacing of the transla
tional levels is very much smaller than kT even when T = 1°K, and
we can safely change the sum for z^ r into an integral over dk, dm,
and dn,
tr = //"/ «P f 4KF + (§) 2 + (ft) 2 ]} dk  «*
= (V/h 3 )(27rMkT) 3/2 [but see Eq. (2113)] (216)
by using Eqs. (126) (note that we have changed from ft back to
h= 27rn).
This result has been obtained by summing over the quantized states
But with the levels so closely spaced we should not have difficulty in
obtaining the same result by integrating over phase space. The trans
lational Hamiltonian is p 2 /2M and the integral is
Ztr = f_l $ exp L2^kT < p x + p y + Pz)] h " 3 dx dy dz dp x d Py d Pz
= (V/h 3 )(27rMkT) 3/2 [but see Eq. (2113)] (217)
184 STATISTICAL MECHANICS
as before. Integration in (217) goes just the same as in (216), ex
cept that we integrate over px,Py,Pz from <*> to + °°, whereas we in
tegrated over k,m,n from to °°; the result is the same.
The probability ffcmn that a molecule has translational quantum
numbers k,m,n is thus (l/z^ r ) exp (ej^n/kT) and the probability
density that a molecule has translational momentum p and is located
at r in V is
f(q,p) = (l/V)(27rMkT) 3/2 exp (p 2 /2MkT)
which is the Maxwell distribution again. Also, in the range of temper
ature where only Z± r changes appreciably with temperature, the
Helmholtz function and the entropy of the gas are
F = kT ln(Z tr ) = NkT Tln(V) + 1 ln(27TMkT/h 2 )l
S = Nk [ln(V) +  ln(27rMkT/h 2 )l + Nk (218)
[but see Eq. (2114)]
There is a ro ijor defect in this pair of formulas. Neither F nor S
satisfies the requirement that it be an extensive variable, as illus
trated in regard to U in the discussion preceding Eq. (63) (see also
the last paragraph in Chapter 8). Keeping intensive variables constant,
increasing the amount of material in the system by a factor A should
increase all extensive variables by the same factor A. If we increase
N to AN in formulas (218), the temperature term will increase by
the factor A but the volume term will become ANk In (AV), which is
not A times Nk ln(V). The corresponding terms in Eqs. (821), giving
the thermodynamic properties of a perfect gas, are Nk ln(V/V ), and
when N changes to AN, V goes to AV and also V goes to AV , so that
the term becomes ANk ln(AV/AV ), which is just A times Nk ln(V/V ).
Evidently the term Nk ln(V) in (218) should be Nk ln(V/N), or some
thing like it, and thus the partition function of (217) should have had
an extra factor N" 1 , or the partition function for the gas should have
had an extra factor N"" N (or something like it). The trouble with the
canonical ensemble for a gas seems to be in the way we set up the
partition function.
If we remember Stirling's formula (185) we might guess that
somehow we should have divided the Z of Eq. (211) by N! to obtain
the correct partition function for the gas. The resolution of this di
lemma, which is another aspect of Gibbs' paradox, mentioned at the
end of Chapter 6, lies in the degree of distinguishability of individual
molecules.
STATISTICAL MECHANICS OF A GAS 185
The Indistinguishability of Molecules
Before the advent of quantum mechanics we somehow imagined that,
in principle, we could distinguish one molecule from another— that we
could paint one blue, for example, so we could always tell which one
was the blue one. This is reflected in our counting of trans lational
states of the gas, for we talked as though we could distinguish between
the state, where molecule 1 has energy e x and molecule 2 has energy
e 2 ,from the state, where molecule 1 has energy e 2 and molecule 2 has
energy e lt for example. But quantum mechanics has taught us that we
cannot so distinguish between molecules; a state where molecule 1 has
quantum numbers k 1 ,m 1 ,h 1 , molecule 2 has k2,m 2 ,n 2 , and so on, not
only has the same energy as the one where we have reshuffled the
quantum numbers among the molecules, it is really the same state,
and should only be counted once, not N! times. We have learned that
physical reality is represented by the wave function, and that the
square of a wave function gives us the probability of presence of a
molecule but does not specify which molecule is present. Different
states correspond to different wave functions, not to different permu
tations of molecules.
At first sight the answer to this whole set of problems would seem
to be to divide Z by N!. If particles are distinguishable, there are N!
different ways in which we can assign N molecules to N different
quantum states. If the molecules are indistinguishable there is only
one state instead of N! ones. This is a goodenough answer for our
present purposes. But the correct answer is not so simple as this, as
we shall indicate briefly here and investigate in detail later. The dif
ficulty is that, for many states of some systems, the N particles are
not distributed among N different quantum states; sometimes several
molecules occupy the same state.
To illustrate the problem, let us consider a system with five par
ticles, each of which can be in quantum state with zero energy or
else in quantum state 1 with energy e. The possible energy levels Ej,
of the system of five particles are, therefore,
all five particles in lower state
one particle in upper state, four in lower
two particles in upper state, three in lower
three particles in upper state, two in lower
four particles in upper state, one in lower
all five particles in upper state
(Note that we must distinguish between system states, with energies
E Pf and particle states, with energies and €.) There is only one
system state with energy E , no matter how we count states. All par
ticles are in the lower particle state and there is no question of which
particle is in which state. In this respect, a particle state is like the
E
=
Et
= e
E 2
= 2e
E 3
= 3e
E 4
= 4e
E R
= 5£
186 STATISTICAL MECHANICS
mathematician's urn, from which he draws balls; ordering of parti
cles inside a single urn has no meaning; they are either in the urn or
not.
Distinguishability does come into the counting of the system states
having energy E lf however. If we can distinguish between particles
we shall have to say that five different system states have energy E 1 ;
one with particle 1 in the upper state and the others all in the lower
"urn," another with particle 2 excited and 1, 3, 4, and 5 in the ground
state, and so on. In other words the multiplicity g x of Eq. (198) is 5
for the system state v  1. On the other hand, if we cannot distinguish
between particles, there is only one state with energy E l5 the one with
one particle excited and four in the lower state (and it has no meaning
to ask which particle is excited; they all are at one time or other, but
only one is excited at a time).
For distinguishable particles, a count of the different ways we can
put five particles into two urns, two in one urn, and three in the other,
will show that the appropriate multiplicity for energy E 2 is g 2 = 10.
And so on; g 3 = 10, g 4 = 5, g 5 = 1. Therefore, for distinguishable par
ticles, the partition function for this simple system would be
Z =
J2 gpe U6/kT = 1 + 5x + 10x 2 + 10x 3 + 5x 4 +
^ =
= (1 + x) 5 where x = e '
and where we have used the binomial theorem to take the last step.
Thus such a partition function factors into single particle factors
z = 1 + e" €//kT , as was assumed in Eqs. (198) and (243).
On the other hand, if the particles are indistinguishable, all the
multiplicities g are unity and
Z = 1 + x + x 2 + x 3 + x 4 + x 5
which does not factor into five singleparticle factors.
Counting the System States
Generalizing, we can say that if we have N distinguishable parti
cles, distributed among M different quantum states, nj of them in
M
particle state j, with energy e^ (so that Y] n i = N), then the number of
J = 1
different ways we can distribute these N particles among" the M par
M
«.. B ,.,„U»,,*.,, te „.n. WE ,. )?i » i « i ,U
STATISTICAL MECHANICS OF A GAS 187
N!, the number of different ways all N particles can be permuted,
being reduced by the numbers nj ! of different ways the particles
could be permuted in each of the M "urns," since permutation inside
an urn does not count. The Z for distinguishable particles then is
z dist = J2 Zv exp ( " Zj nj 6j/kT
= V — x ni x n2 ••• x nM = z N (2110)
v
where xi = exp (e^/kT), z = x x + x 2 + ••• x^, where the sum is over
J M
all values of the m's for which V n i = N, and where we have used
j=i J
the multinomial theorem to make the last step. Again this partition
function factors into singleparticle factors.
Again, if the particles are indistinguishable, the partition function
is
Z ind
;rx^x n M (21n)
with the sum again over all values of the nj's for which £nj  N.
This sum does not factor into singleparticle factors.
We thus have reached a basic difficulty with the canonical ensem
ble. As long as we could consider the particles in the gas as distin
guishable, our partition functions came out in a form that could be fac
tored into N z's, one for each separate particle. As we have seen,
this makes the calculations relatively simple. If we now have to use
the canonical ensemble for indistinguishable particles, this factorabil
ity is no longer possible, and the calculations become much more dif
ficult. In later chapters we shall find that a more general ensemble
enables us to deal with indistinguishable particles nearly as easily as
with distinguishable ones. But in this chapter we are investigating
whether, under some circumstances, the partition function for the
canonical ensemble can be modified so that indistinguishability can
approximately be taken into account, still retaining the factorability
we have found so useful. Can we divide Z^^ of Eq. (2110) by some
single factor so it is, at least" approximately, equal to the Z mc j of
Eq. (2111)?
188 STATISTICAL MECHANICS
There is a large number of terms in the sum of (2110) which have
multiplicity g^ = N! These are the ones for the system states v, for
which all the nj's are or 1, for which no particle state is occupied
by more than one particle. We shall call these system states the
sparse states, since the possible particle states are sparsely occu
pied. On the other hand, there are other terms in (2110) with multi
plicity less than N!. These are the terms for which one or more of
the n^'s are larger than 1; some particle states are occupied by more
than one of the particles. Such system states can be called dense
states, for some particle states are densely occupied. If the number
and magnitude of the terms for the sparse states in (2110) are much
larger than the number and magnitude of the terms for the dense
states, then it will not be a bad approximation to say that all the gj/s
in (2110) are equal to N! and thus that (Zdist/ N  f ) does not differ
much from the correct Z^. And (Zdist/N!) can still be factored,
although Zi n d cannot.
To see when this advantageous situation will occur, we should ex
amine the relative sizes of the terms in the sum of Eq. (2110). The
term for which the factor x 1 1 ••• x ^ is largest is the one for which
n x = N, nj = (j > 1) (i.e., for which all particles are in the lowest
state). This term has the value 1 • exp (Nc^kT). It is one of the
"densest" states. The largest term for a sparse state is the one for
which n x = n 2 = ••• = n^ = 1, nj = (j >N) (i.e., for which one particle
is in the lowest state, one in the next, and so on up to the Nth state).
Its value is
(N!) exp [(€,+ e 2 + ••• + e N )/kT]
« V2ttN exp (N[ln(N/e)  (e N /kT)]}
where we have used Stirling's formula (185) for N! and we have writ
ten e~N for the average energy [(e 1 + e 2 + ••• + €n)/N] of the first N par
ticle states. Consequently, whenever ln(N/e) is considerably larger
than (e"jyf  e 1 )/kT, the sum of sparsestate terms in Zdist is so much
larger than the sum of densestate terms that Zdist is practically
equal to a sum of the sparsestate terms only, and in this case Zdist
ea NlZind* Tn i s situation is the case when kT is considerably larger
than the spacing between particlestate energy levels, which is the
case when classical mechanics holds.
The Classical Correction Factor
Therefore whenever the individual particles in the system have en
ergy levels sufficiently closely packed, compared to kT, so that clas
sical phasespace integrals can be used for at least part of the z
STATISTICAL MECHANICS OF A GAS 189
factor, it will be a good approximation to correct for the lack of dis
tinguishability of the molecules by dividing Zdist by N ! . In this case
there are enough lowlying levels so that each particle can occupy a
different quantum state and our initial impulse, to divide Z by N!,
the number of different ways in which we can assign N molecules to
N different states, was a good one. Instead of Eq. (211) we can use
the approximate formula
Z  (l/N!)(z mo ie) N  (ez m0 le/N) N (2112)
(omitting the factor V27iN in the second form).
Since the translational energy levels of a gas are so closely spaced,
this method of correcting for the indistinguishability of the molecules
should be valid for T > 1°K.
The correction factor can be included in the translational factor, so
that, instead of Eqs. (214) to (218), we should use
Z tr = (l/N!)V N (27TMkT/h 2 ) (3/2)N ^(eV/N) N (27rMkT/h 2 ) (3/2)N
 (eV/n£ 3 T ) N (2113)
where n = N/N is th e numbe r of moles and where the "thermal
length" i T = hNj /3 /V27rMkT is equal to 1.47 x 10" 2 meters for pro
tons at T = 1°K (for other molecules or temperatures divide by the
square root of the molecular weight or of T). The values of the trans
lational parts of the various thermodynamic quantities for the gas,
corrected for molecular indistinguishability, are then
F tr = NkT["ln(eV/N) +  In (27iMkT/h 2 ) 1
Str = Nk[ln (V/N) +  In (27rMkT/h 2 )l +  Nk
U = NkT; C v = Nk; P = (NkT/V)
H = NkT; C p =Nk (2114)
The equation for S is called the Sac kurTetrode formula.
Comparison with Eqs. (821) shows that statistical mechanics has
indeed predicted the thermodynamic properties of a perfect gas. It
has done more, however; it has given the value of the constants of in
tegration S , T , and V in terms of the atomic constants h, M, and k,
and it has indicated the conditions under which a collection of N mol
ecules can behave like a perfect gas of point particles.
190 STATISTICAL MECHANICS
We also discover that we can now solve Gibbs' paradox, stated at
the end of Chapter 6. Mixing two different gases does change the en
tropy by the amount given in Eq. (614). But mixing together two por
tions of the same gas produces no change in entropy. If the molecules
on both sides of the diaphragm are identical, there is really no in
crease in disorder after the diaphragm is removed. One can never
tell (in fact one must never even ask) from which side of the diaphragm
a given molecule came, so one cannot say that the two collections of
identical molecules "intermixed" after the diaphragm was removed.
We also note that division by N! was not required for the crystal
discussed in Chapter 20. In a manner of speaking, N! was already
divided out. We never tried to include, in our count, the number of
ways the N atoms could be assigned to the different lattice points, and
so we did not have to divide out the number again. More will be said
about this in Chapter 27.
The Effects of Molecular Interaction
We have shown several times [see Eqs. (179), (1819), and (2114)]
that, when the interaction between separate molecules in a gas is ne
glected completely, the resulting equation of state is that of a perfect
gas. Before we finish discussing the translational partition function
for a gas, we should show how the effects of molecular interaction can
be taken into account. We shall confine our discussion to modifications
of the translational terms, since these are the most affected. Molecu
lar interactions do change the rotational, vibrational, and electronic
motions of each molecule, but the effects are smaller.
The first effect of molecular interactions is to destroy the factor
ability of the translational partition function, at least partly. The
translational energy, instead of being solely dependent on the molecu
lar momenta, now has a potential energy term, dependent on the rela
tive positions of the various molecules. This is a sum of terms, one
for each pair of molecules. The force of interaction between molecule
i and molecule j, to the first approximation, depends only on the dis
tance ry between their centers of mass. It is zero when ry is large;
as the molecules come closer together than their average distance the
force is first weakly attractive until, at r^ equal to twice the "radius"
r of each molecule, they "collide" and their closer approach is pre
vented by a strong repulsive force. Thus the potential energy Wij(rij)
of interaction between molecule i and molecule j has the form shown
in Fig. 211, with a small positive slope (attractive force) for r^ >
2r and a large negative slope (repulsive force) for rij < 2r . By the
time rij is as large as the average distance between molecules in the
gas, Wy is zero; in other words we still are assuming that the ma
jority of the time the molecules do not affect each other.
The translational part of the Hamiltonian of the system is
STATISTICAL MECHANICS OF A GAS
191
Fig. 211. Potential energy of interaction between two mole
cules as a function of their distance apart.
N
H tr = (1/2M) Z] Pi + La W ij( r ij)
i = 1 all pairs
(2115)
where the sum of the Wy's is over all the (1/2)N(N 1) « (1/2JN 2
pairs of molecules in the gas. The translational partition function
is then
z tr = / — /e~ tr dxi dx 2 ••• dz N dp x l dp y i ••• dp Z N/h
The integration over the momentum coordinates can be carried
through as with Eq. (217) and, since the molecules are indistin
guishable, we divide the result by N!. However the integration over
the position coordinates is not just V^ this time, because of the pres
ence of the Wij's,
1 /2*MkT\ (3 / 2)N
Ztr  Zp * Zq ;
Z P^N!
h 2 )
e\ N /2,MkT\ (3 / 2)N
z a = /—/exp
N/ \ h 2 )
 H WytryJAT
_ all pairs
dx x dy x ••• dy N dz N
(2116)
Let us look at the behavior of the integrand for Zq, as a function of
192 STATISTICAL MECHANICS
the coordinates x 1 ,y 1 ,z 1 of one molecule. The range of integration is
over the volume V of the container. Over the great majority of this
volume the molecule will be far enough away from all other molecules
so that £Wij is and the exponential is 1; only when molecule 1
comes close to another molecule (the jth one, say) does rjj become
small enough for Wjj to differ appreciably from zero. Of course if
rH becomes smaller than 2r , W]i becomes very large positive and
the integrand for Zq will vanish. The chance that two molecules get
closer together than 2r is quite small.
Thus it is useful to add and subtract 1 from the integrand,
Zq = / "'/{I + [exp (£Wij/kT) _ x]} dXi ... dzN
= V N + /../[expt^Wij/kT) 1] dx,." dz N (2117)
where the first unity in the braces can be integrated as in Eq. (217)
and the second term is a correction to the perfect gas partition func
tion, to take molecular interaction approximately into account. As we
have just been showing, over most of the range of the position coordi
nates the integrand of this correction term is zero. Only when one of
the r^j's is relatively small is any of the W^j's different from zero.
To the first approximation, we can assume that only one Wjj differs
from zero at a time, as the integration takes place.
Thus the integral becomes a sum of similar integrals, one for each
of the (1/2)N(N 1) ^ (1/2)N 2 interaction terms W^. A typical one is
the integral for which Wij is not zero; for this one the integrand dif
fers from zero only when (x 1 ,y 1 ,z 1 ) is near (xj,yj,Zj), so in the integra
tion over dK x dy x dz x = dV\ we could use the relative coordinates
rij,#lj,01j. Once this integral is carried out, the integrand for the
rest of the integrations is constant, so each of the integrals over the
other dVi's is equal to V. Thus
Z q = V N + I N* fd^u / sin fli] dflx, /( e  W lj/ kT  l)r 2 Xj drjj
*/ /dv, dV N
^tiffv"" 1
^/(eWlJ^l^drJ
When r^j < 2r , Wij becomes very large positive and the integrand
of the last term becomes 1, so this part of the quantity in brackets
is just minus the volume of a sphere of radius 2r , which we shall
call 2/3. For rjj > 2r , Wij is small and negative, so (e~ W lj/ kT  1)
 (Wij/kT), and this part of the quantity in brackets is roughly
STATISTICAL MECHANICS OF A GAS 193
oo
4tt / (Wij/kT) rj drij
2r o
which we shall call 2a /kT.
The Van der Waals Equation of State
Therefore, to the first approximation, molecular interaction
changes Ztr from the simple expression of Eq. (2113) to
where Nj3 = N(87rr£/3) is proportional to the total part of the volume V
which is made unavailable to a molecule because of the presence of the
other molecules, and where a? is a measure of the attractive potential
surrounding each molecule. The /3 and a. terms in the bracket are
both small compared to 1.
The Helmholtz function and the entropy for this partition function
are
F tr    NkT l„pfl)N k T l„(f )« in (l ^ + ^)
. NkTln (^) NkTln (f) + NkT (f)_(^)
— NHln(*pr
)«" [f(f)](^)
S tr  §Nk + §Nk ln(^») + Nk In [(V N0)] (2119)
Comparison with Eqs. (2114) shows that U and C v are unchanged,
to this approximation, by the introduction of molecular interaction.
However the equation of state becomes
\dVlm ~ V 
NkT N 2 a
T V  N/3 V
or
(p + ^r)(V  Nj3) =* NkT = nRT (2120)
which is the Van der Waals equation of state of Eq. (34), with
a = N*,a and b = N /3. The correction N 2 a/V 2 to p (which tends to
decrease P for a given V and T) is caused by the small mutual
194 STATISTICAL MECHANICS
attractions between molecules; the correction N/3 to V (which tends
to increase P for a given V and T) is the volume excluded by the
presence of the other molecules. Thus measurement of a and b from
the empirical equation of state can give us clues to molecular sizes
and attractive forces; or else computation of the forces between like
molecules can enable us to predict the Van der Waals equation of
state that a gas of these molecules should obey.
A Gas of
Diatomic
Molecules
In the molecular gas described in the preceding chapter, as long as
kT is small compared to the energy spacing of rotational quantum lev
els of individual molecules, only Ztr differs appreciably from unity and
the gas behaves like a perfect gas of point atoms (if we neglect molecu
lar interactions). To see for what temperature range this holds, we
need to know the expression for the allowed energies of free rotation
of a molecule. This expression is quite complicated for polyatomic
molecules, so we shall go into detail only for diatomic molecules.
The Rotational Factor
If the two constituent nuclei have masses M x and M^ and if they
are held a distance R apart at equilibrium, the moment of inertia of
the molecule, for rotation about an axis perpendicular to R through
the center of mass, is I = [M^gR 2 ,/^! + M 2 )] . The moment of inertia
about the R axis is zero. The kinetic energy of rotation is then 1/21
times the square of the total angular momentum of the molecule.
This angular momentum is quantized, of course, the allowed values
of its square being "h 2 £(£ + 1), where I is the rotational quantum num
ber, and the allowed values of the component along some fixed direc
tion in space are one of the (2L+ 1) values £li, (£  l)fi, ... , +
(i  l)h, +IH, for each value of I. Put another way, there are 2£ + 1
different rotational states which have the energy (*h 2 /2I)£U + 1), so the
partition function for the rotational states of the gas system of N mol
ecules is
f 2 "IN
Zrot = l Z] (2£+1) ex PMrot* U+1 )/T]l (22D
where # ro t = ti 2 /2Ik. Therefore when T is very small compared to
0rob z rot  1 and > according to the discussion following Eq. (1911),
the rotational entropy and specific heat are negligible.
195
196 STATISTICAL MECHANICS
Values of # ro t f° r a * ew diatomic molecules will indicate at what
temperatures Z ro t begins to be important. For H 2 , # ro t = 85 °K; for
HD, e T0t = 64°K; for D 2 , £ ro t = 47 °K; for HC1, ro t = 15°K; and for
2 , ro t  2°K. Therefore, except for protium (hydrogen), protium
deuteride, and deuterium gases, T is appreciably larger than ro t in
the temperature range where the system is a gas.
In these higher ranges of temperature we can change the sum for
Z into an integral,
_ _ , ,N
z rot ~ ^ z voV
1
 f(2l+ 1) exp[6 rot (£ 2 +£)/T] di = T/6 TOt
so
Zrot * (T/^rot) N = (87T 2 IkT/h 2 ) N
F rot  NkT In (T/0 rot ); S rot  Nk ln(eT/0 rot )
U rot NkT; C^ 0t Nk, T»0 rot (222)
Thus for a gas of diatomic molecules at moderate temperatures,
where both translational and rotational partition functions have their
classical values, the total internal energy is (5/2)NkT and the total
heat capacity is (5/2)Nk, as mentioned in the discussion following Eq.
(1311). The rotational terms add nothing to the equation of state,
however, for the effect of the neighboring molecules on a molecule's
rotational states is negligible for a gas of moderate or low densities;
consequently Z ro t and F ro t are independent of V. Therefore the
equation of state is determined entirely by Zt r , unless the gas density
is so great that not even the Van der Waals equation of state is valid.
For hydrogen and deuterium, a more careful evaluation of Eq.
(221) results in
z rot~* ^
1 + 3e 2£rot/T\ N , T<6 TOt
Ng rot /4T7 t + \_ + ie rot
\6 T0t 12 480T / ' ^ rot
rot
A plot of the exact value of C v /Nk, plotted against T/0 rot is shown
rot
in Fig. 221. We note that C v rises somewhat above Nk = nR, as T
increases, before it settles down to its classical value. The measured
A GAS OF DIATOMIC MOLECULES
197
Fig. 221. The rotational part of the heat capacity of a di
atomic gas as a function of temperature.
values of Cy 0t for HD fit this curve very well, from T = 35 °K to sev
eral hundred degrees K, when molecular vibration begins to make it
self felt. But the C£ ot curves (i.e., C v  cty) for H 2 and D 2 do not
match, no matter how one juggles the assumed values of 6 ro i; f° r ex ~
ample, the curve for H 2 has no range of T for which C$ ot > Nk, and
the peak for D 2 is not as large as Fig. 221 would predict. The expla
nation of this anomaly lies again with the effects of indistinguis liability
of particles. The hydrogen and deuterium homonuclear molecules, H 2
and D 2 , are the only ones with a lowenough boiling point so that these
effects can be measured. The effects would not be expected for HD,
for here the two nuclei in the molecule differ and are thus distinguish
able. The calculations for H 2 and D 2 will be discussed later, in Chap
ter 27, after we take up in detail the effects of indistinguishability.
The Gas at Moderate Temperatures
Therefore, for all gases except H 2 , HD, and D 2 , over the temper
ature range from the boiling point of the gas to the temperature v i D >
where vibrational effects begin to be noticeable, the only effective fac
tors in the partition function are those for translation and rotation,
and these factors can be computed classically, using Eqs. (2114) and
(222). In this range we can also calculate the partition function for
polyatomic molecules. The classical Hamiltonian is (Pi/21^ + (p 2 /2I 2 )
+ (P3/2I3), where l lf I 2 , I3 are the moments of inertia of the molecule
about its three principle axes and p x , p 2 , p 3 are the corresponding an
gular momenta. Therefore,
198 statistical mechanics
, [(p!A) + (pI/i 2 ) + (
Z rot * 1 (877 2 /h 3 ) J J J exp
J rot " V /**> J J J «■* \ 2kT
N
x dpj dp 2 dp,
= [(8ir 2 /ah 3 )VirT^ (27ikT) 3/2 ] N (224)
where Su 2 is the factor produced by the integration over the angles
conjugate to Pi,p 2 >P3 and where a is a symmetry factor, which enters
when two or more indistinguishable nuclei are present in a molecule.
(If the molecule is asymmetric, cr= 1; if it has one plane of symmetry,
a = 2; etc.)
We can now write the thermodynamic functions for a gas for which
molecular interactions are negligible, for the temperature range where
kT is large compared with rotationalenergylevel differences but
small compared with the vibrational energy spacing;
For monatomic gases, there is no Z ro t and, from Eq. (2114),
F « F tr =* NkT
ln(V/N) + lnT + F J
U^NkT; C v ^Nk; P « NkT/V
For diatomic gases, use Eq. (222) for Z ro t, and
F  F tr + F rot  NkT
ln(V/N) +  In T + F 1
U « NkT; C v « Nk; P « NkT/V (225)
For polyatomic gases, use Eq. (224) for Z ro t, and
F « Ftp + F rot « NkT [in (V/N) + 3 In T + F ]
U * 3NkT; C v « 3Nk; P « NkT/V
where the constant F is a logarithmic function of k, h, the mass M
of the molecule, and of its moments of inertia, the value of which can
be computed from Eqs. (2114) and (222) or (224). All these for
mulas are for perfect gases, in that the equation of state is PV = NkT
and the internal energy U is a function of T only. The specific heats
depend on the nature of the molecule, whether it is monatomic, di
atomic, or polyatomic.
A GAS OF DIATOMIC MOLECULES
199
We note that the result corresponds to the classical equipartition
of energy for translational and rotational motion, U being (l/2)kT
times the number of ' 'unfrozen" degrees of translational and rota
tional freedom. The effects of molecular interaction can be allowed
for approximately by adding the factor in brackets in Eq. (2118) to Z.
These results check quite well with the experimental measurements,
mentioned following Eq. (1311).
The Vibrational Factor
When the temperature is high enough so that kT begins to equal
the spacing between vibrational levels of the molecules, then Z Y ^ be
gins to depend on T and the vibrational degrees of freedom begin to
"thaw out." In diatomic molecules there is just one such degree of
freedom, the distance R between the nuclei. The corresponding poten
tial energy W(R) has its minimum value at R , the equilibrium sep
aration between the nuclei, and has a shape roughly like that shown in
Fig. 222. As R — «>, the molecule dissociates into separate atoms;
R — ^
Fig. 222. Diatomic molecular energy W(R) as a function
of the separation R between nuclei.
the energy required to dissociate a molecule from equilibrium is D,
the dissociation energy.
If the molecule is rotating there will be added a dynamic potential,
corresponding to the centrifugal force, which is proportional to the
square of the molecule's angular momentum and inversely proportional
200 STATISTICAL MECHANICS
to R 3 . Fortunately, for most diatomic molecules, this term, which
would couple Z ro t and Z v ifc, is small enough so we can neglect it
here. For small amplitude vibrations about R the system acts like
a harmonic oscillator, with a natural frequency u)/2n which is a func
tion of the nuclear masses and of the curvature of the W(R) curve
near R . Thus the lower energy levels are "nu>(n + 1/2), where n is
the vibrational quantum number.
Therefore, to the degree of approximation which neglects coupling
between rotation and vibration and which considers all the vibrational
levels to be those of a harmonic oscillator,
z vib ~
[eV^J
\iN
,lWkTV
(226)
where e = W(R ) + (l/2)fico, and where we have used Eq. (202) to re
duce the sum. The corresponding contributions to the Helmholtz func
tion, entropy, etc., of the gas are
kT<Hw
kT>1ia>
F vib^
Ne + NkT lnfl  e
*w/kT\
u vib
Ne +
e +liwAT _ x
fNe + Nna;e^ /k
[Ne + NkT
r vib ~
NfiV
kT 2
e Uw/kT
/1iw/kT _
0'
{
(Nn 2 a> 2 /kT 2 ) e
Nk
■Bw/kT
kT <*Ctfw
kT >Hco
(227)
which are added to the functions of Eqs. (225) whenever the tempera
ture is high enough (for T equal to or larger than 6 v fo ="nu>/k).
As examples of the limits above which these terms become appre
ciable, the quantity vib is equal to 2200°K for 2 , to 4100°Kfor
HC1 and to 6100°K for H 2 (Fig. 223). Therefore below roughly
1000°K the contribution of molecular vibration to S, U, and C v of
diatomic gases is small. Above several thousand degrees, the vibra
tional degree of freedom becomes "unfrozen," an additional energy
kT is added per molecule, and an additional Nk to C v [a degree of
A GAS OF DIATOMIC MOLECULES
nR
201
>>
Fig. 223. Vibrational part of the heat capacity of a di
atomic molecular gas as a function of temper
ature.
freedom with quadratic potential classically has energy kT; see
Eq. (1314)].
In the case of a polyatomic molecule with n nuclei, there are
3n 6 vibrational degrees of freedom, each with its fundamental
frequency u>j/27T. The vibrational partition function is
Ne
^vib _ e
z r ( l . e  1l V kT )
Z X Z 2 ... Z 3n _ 6
N
where
(228)
[compare this with Eq. (207)] . Again, for polyatomic gases, the vi
brational contribution below about 1000°K is small, at higher T the
contribution to U is N(3n 6)kT. It often happens that the molecules
are dissociated into their constituent atoms before the temperature is
high enough for the vibrational term to "unfreeze."
The temperature would have to be still higher before Z e \ began to
have any effect. The usual electronic level separation divided by k is
roughly equal to 10,000 °K at which temperatures most gases are dis
sociated and partly ionized. Such cases are important in the study of
stellar interiors, but are too complex to discuss in this book. And be
fore we can discuss the thermal properties of electrons we must re
turn to first principles again.
The Grand
Canonical
Ensemble
The canonical ensemble, representing a system of N particles kept
at constant temperature T, has proved to be a useful model for such
systems as the simple crystal and the perfect (or nearly perfect) gas.
Many other systems, more complicated than these, can also be repre
sented by the canonical ensemble, which makes it possible to express
their thermodynamic properties in terms of their atomic structure.
But in Chapter 21 we discovered a major defect, not in the accuracy
of the canonical ensemble when correctly applied, but in its ease of
manipulation in some important cases.
Whenever the N particles making up the system are identical and
indistinguishable, the corresponding change in the multiplicity factors
g v has the result that the correct partition function does not separate
into a product of N independent factors, even if the interaction be
tween particles is negligible. In cases where kT is large compared
to the separation between quantum levels of the system, we found we
could take this effect into account approximately by dividing by N!. In
this chapter we shall discuss a more general kind of ensemble, which
will allow us to retain f actorability of partition function and at the
same time take indistinguis liability into account exactly, no matter
what value T has.
An Ensemble with Variable N
The new ensemble, which we shall call the grand canonical ensem
ble, is one in which we relax the requirement that we placed on the
microcanonical and canonical ensembles— that each system in the en
semble has exactly N particles. We can imagine an infinitely large,
homogeneous supersystem kept at constant T and P. The system the
new ensemble will represent is that part of the supersystem contained
within a volume V. We can imagine obtaining one of the sample sys
tems of the ensemble by withdrawing that part of the supersystem
which happens to be in a volume V at the instant of removal, and of
202
THE GRAND CANONICAL ENSEMBLE 203
doing this successively to obtain all the samples that make up the en
semble. Not only will each of the samples differ somewhat in regard
to their total energy, but the number of particles N in each sample
will differ from sample to sample. Only the average energy U and the
average number of particles N, averaged over the ensemble, will be
specified.
The equations and subsidiary conditions serving to determine the
distribution function are thus still more relaxed than for the canonical
ensemble. A microstate of the grand canonical ensemble is specified
by the number of particles N that the sample system has, and by the
quantum numbers i?jj = u lf v 2 , ... , ^3^, which the sample may have
and which will specify its energy E N ^. Thus for an equilibrium mac
rostate the distribution function f$ v must satisfy the following re
quirements:
S = k X! f N ln f N^ is maximum
subject to
L fN^ =1 ; L E Nl/ f N ^ = U; Z>f N „ = N
> (23D
where U, N, and S are related by the usual thermodynamic relation
ships, such as U = TS + SI + N/i, for example, or any other of the
equations (821). Function £2 is the grand potential of Eq. (815). Note
that, jnstead of n, the mean number of moles in the system, we now
use N, the mean number of particles, and therefore ju. is now the
chemical potential (the Gibbs function) per particle, rather than per
mole, as it was in the first third of this book. We shall consistently
use it thus henceforth, so it should not be confusing to use the same
symbol, /1.
The Grand Partition Function
As before, we simplify the requirements by using Lagrange multi
pliers, and require that
~ k E f N^ lnfNiz + af! 2] f Ni , + a e £ E Nl ,f Nl ,
N,i> N,i> N,y
+ a n Zj Nf^j, be maximum, (232)
with a l9 a e , a n chosen so that
204 STATISTICAL MECHANICS
E f N^ =1 ; E En^N^U; E Nf Nj , = N (232)
N,^ N,z^ N,v
The partials with respect to the fNi/ s > which must be made zero, re
sult in the equations
k In f Nj , + k = a 1 + a n N + a e E Nl ,
or
f N „ = exp[(l/k)(a 1 k + a n H + a n E Nl ,)] (233)
The first requirement is met by setting
e (k Q ' 1 )/k = ^ = J^ e (a e N + a e E Np )/k
The other two are met by inserting this into the expression for S,
S =  E f Ni>( Q 'i"" k + a n N + «e E Nt>) = (k o^)  a n N a e U
N,y
and then identifying this with the equations S = (U  J2  Njll)/T, from
Eq. (821).
We see that we must have
ka 1 = (Sl/T) = k In 5; ot n = M/T; a e = (1/T)
so that the solution of Eq. (231) is
*N„ = (V3) exp ^ — — J ; 3 = /_> exp ^ — j^— J
12 = kT In 3 = PV; (aft/a^iV = N
OJ2/3T) VjUL = S; 0O/8V) T/1 = P (234)
C v = T(aS/8T) V/i ; F = ft + /iN; U = F + ST = ft + ST+jiiN
These are the equations for the grand canonical ensemble. The
sum ? is called the grand partition function; it is the sum of the ca
nonical partition functions Z(N) for ensembles with different N's,
with weighting factors e^N/kT^
THE GRAND CANONICAL ENSEMBLE 205
3 = £ e^ N/kT Z(N); Z(N) = £ e" E ^/ kT (235)
N = v
All the thermodynamic properties of the system can be obtained from
£1 by differentiation, as with the canonical ensemble. We shall see that
this partition function has even greater possibilities for factoring than
does its canonical counterpart.
The Perfect Gas Once More
Just to show how this ensemble works we take up again the familiar
theme of the perfect gas of point particles. From Eq. (2113) we see
that, if we take particle indistinguishability approximately into ac
count, the canonical partition function for the gas of N particles is
,N
Z(N) « (l/N!)(V/je) ; t t = (h/V2?rMkT
and therefore the grand partition function is, from Eq. (235),
N
N =
(l/N!)[(V/£?)e M/kT ] =exp [(V/£j)e /i/kT ] (236)
where we have used the series expression e = £] (x /n!).
Then, from Eqs. (234) n =
Q= kTV(27rMkT/h 2 ) 3/2 e M//kT = PV
N = V(277MkT/h 2 ) 3/2 e M//kT = (J2/kT) = PV/kT
S = kV(277MkT/h 2 )3/ 2 e /i/kT ( >^) = Nk(  ^)
U = S2+ST + /iN=NkT + NkT(  ^) + /iN = fikT
F = N(jukT); jll = kT ln[(V/N)(27rMkT/h 2 ) 3 / 2 ] (237)
which of course present, in slightly different form, the same expres
sions for U, C v , and the equation of state as did the other ensembles;
only now N occurs instead of N. We also obtain directly an expres
sion for the chemical potential per particle, jul, for the perfect gas.
The probability density that a volume V of such a gas, in equilib
rium at temperature T and chemical potential /i, happens to contain
206
STATISTICAL MECHANICS
N particles, and that these particles should have momenta p x p 2 , ... ,
PN and be located at the points specified by the vectors r lf r 2 , ... , r^ ,
is then
fN^'P)
3N
N! / exp ikT
N
p.N
f— »\2M/
_ v / 27rMkT \ 3/2 e jLiAT
(238)
This is a generalization of the Maxwell distribution. The expression
not only gives us the distribution in momentum of the N particles
which happen to be in volume V at that instant (it is of course inde
pendent of their positions in V), but it also predicts the probability
that there will be N molecules in volume V then. If we should wish
to use P and T to specify the equilibrium state, instead of \i and T,
this probability density would become
%(q,p)
(P/kT)
N
N! (2*MkT)( 3 /2)N
exp
N
L
3 Pv
2MkT
(239)
Density Fluctuations in a Gas
By summing f^ over v for a given N (or by integrating in(cup)
over the q's and p's for a given N) we shall obtain the probability
that a volume V of the gas, at equilibrium at pressure P and temper
ature T, will happen to have N molecules in it. From Eqs. (234)
and (235) this is
fN = E f N^ = < 1 /?)e
^ kT Z(N) = e ( ^ + ^ N) / kT Z(N)
Using the expressions for Z(N) and those for (£2/kT) and (ju/kT), we
obtain
N
N
fN\ pf__f 2 ] N (±\
\V) \2nMkTj \N!/
277MkT \ 3/2
h 2 )
nN
= (N N /N!)e"
N =(l/N!)(PV/kT) N e pV / kT
(2310)
This is a Poisson distribution [see Eq. (115)] for the number of
particles in a volume V of the gas. The mean number of particles is
N = PV/kT and the probability % is greatest for N near N in value.
But fjsj is not zero when N differs from N; it is perfectly possible to
find a volume V in the gas which has a greater or smaller number of
molecules in it than PV/kT. The variance of the number present is
THE GRAND CANONICAL ENSEMBLE 207
00 OO
(AN) 2 = Y] (NN) 2 f N  V N 2 f N (N) 2 N = PV/kT
N = N = (2311)
and the fractional deviation from the mean is
AN/N = /N = 1/AT/PV (2312)
(It should be remembered that the system described by the grand ca
nonical ensemble is not a gas of N molecules in a volume V, but that
part of a supersystem which happens to be in a volume V, where V
is much smaller than the volume occupied by the supersystem; thus
the number of particles N that might be present can vary from zero
to practically infinity.)
The smaller the volume of the gas looked at (the smaller the value
of N) the greater is this fractional fluctuation of number of particles
present (or of density, for AN/N = Ap/p). Thus we have arrived at
the result of Eq. (156), for the density fluctuations in various por
tions of a gas, by a quite different route.
Quantum
Statistics
But we still have not demonstrated the full utility of the grand ca
nonical ensemble for handling calculations involving indistinguishable
particles. The example in the previous chapter used the approximate
correction factor (1/N!), which we saw in Chapter 21 was not valid at
low temperatures or high densities. We must now complete the dis
cussion of the counting of states, which was begun there.
Occupation Numbers
In comparing the partition functions for distinguishable and indis
tinguishable particles, given in Eqs. (2110) and (2111) for the canon
ical ensemble, we saw that it was easier to compare the two if we
talked about the number of particles occupying a given particle state
rather than talking about which particle is in which state. In fact if
the particles are indistinguishable it makes no sense to talk about
which particle is in which state. We were there forced to describe the
system state v by specifying the number nj of particles which occupy
the jth particle state, each of them having energy ej. The number nj
are called occupation numbers.
Of course if the interaction between particles is strong (as is the
case with a crystal) we cannot talk about separate particle states; oc
cupation numbers lose their specific meaning and we have to talk about
normal modes instead of particles. But let us start with the particle
interactions being small enough so we can talk about particle states
and their occupation numbers. The results we obtain will turn out to be
capable of extension to the stronginteraction case.
We thus assume that, in the system of N particles, it makes sense
to talk about the various quantum states of an individual particle, which
we call particle states. These states are ranked in order of increasing
energy, so that if ej is the energy of a particle in state j, then £j + l
> ej. Instead of specifying the system state v by listing what state
particle 1 is in, and so on for each particle, we specify it by saying
208
QUANTUM STATISTICS 209
how many particles are in state j (i.e., by specifying nj). Thus when
the system is in state v = (n^, ... , nj, ...), the total number of par
ticles and the total energy are
j j
For the canonical ensemble, we have to construct the partition
function Z for a system with exactly N particles; the sum over
v includes only those values of the nj's for which their sum comes
out to equal N. This restriction makes the calculation of a partition
function like that of Eq. (2111) more difficult than it needs to be.
With the grand canonical ensemble the limitation to a specific value of
N is removed and the summation can be carried out over all the oc
cupation numbers with no hampering restriction regarding their total
sum.
Thus the grand partition function can be written in a form analogous
to the Z of Eqs. (198) and (2110),
5 = ■ £ g v exp [(1/kT) £ nj (/i  €])] (242)
by virtue of Eqs. (241). The multiplicities g u , for each system state
v (i.e., for each different set of occupation numbers nj) are chosen
according to the degree of distinguishability of the particles involved.
Indeed, this way of writing Q, is appropriate also when the "parti
cles" are identical subsystems, such as the molecules of a gas. In
such cases the "particle states" j are the molecular quantum states,
specified by their trans lational, rotational, vibrational, and electronic
tr rot
quantum numbers, the allowed energies €j are the sums €k mn + e^ u
+ e n + e of Chapter 21, and the nj's are the number of molecules
that have the same totality of quantum numbers j = k,m,n,A.,^,n, etc.
However, we shall postpone discussion of this generalization until
Chapter 27.
Maxwell Boltzmann Particles
At present we wish to utilize the grand canonical ensemble to in
vestigate systems of "elementary" particles, such as electrons or
protons or photons or the like, sufficiently separated so that their mu
tual interactions are negligible. Each particle in such a system will
have the same mass m and will be subject to the same conservative
forces, so that the Schrodinger equation for each will be the same.
Therefore, the total set of allowed quantum numbers, represented by
the index j, will be the same for each particle (although at any instant
210
STATISTICAL MECHANICS
different particles may have different values of j). The allowed en
ergy corresponding to the set of quantum numbers represented by j
is ej and the number of particles in this state is nj. The grand parti
tion function can then be written as in Eq. (242).
The values of the multiplicities g p must now be determined for
each kind of fundamental particle. This to some extent is determined
by the nature of the forces acting on the particles. For example, if
the particle has a spin and a corresponding magnetic moment, if no
magnetic field is present the several allowed orientations of spin will
have the same energy, and the g's will reflect this fact. Let us avoid
this complication at first, and assume that each particle state j has
an energy ej which differs from that of any other particle state.
When the elementary particles are distinguishable, the discussion
leading up to Eq. (219) indicates that g Vf the number of different ways
N particles can be assigned to the various particle states, nj of them
being in the jth particle state, is [Nl/n^n^ •••]. Since 0!= 1, we
can consider all the n's as being represented in the denominator, even
those for states unoccupied; an infinite product of l's is still 1. The
grand partition function will thus have the same kind of terms as in
Eq. (2110), but the restriction on the summation is now removed; all
values of the nj's are allowed. Therefore, for distinguishable parti
cles,
3d
ist
L
n x ,n 2 , .
(n x + n 2 +
n x ! n 2 !
)!
exp
l^£ n J (M
J
ei)
(243)
it reduces to the sum
N
In contrast to Eq. (2010), this sum is not separable into a simple
product of oneparticle factors; because of the lack of limitation on N
£ e j , each term of which is
a product of oneparticle partition functions.
In the classical limit, when there are many particle states in the
range of energy equal to kT, the chance of two particles being in the
same particle state is vanishingly small, and the preponderating terms
in series (243) are those for which no m is larger than 1. In this
case we may correct for the indistinguishability of particle by the
''shotgun" procedure, used in Chapter 21, of dividing every term by
the total number of ways in which N particles can be arranged in N
different states. The resulting partition function
? MB
L
n!,n 2 ,
1
n x ! n 2 !
I ...
exp
(l/kT)^(Mej)
QUANTUM STATISTICS
211
n^/xeJ/kT V J_ n 2 (pe 2 )/kT
= exp
E
 3
(MCj)AT
?i^
*J
exp e
(Mcj)/kT
(244)
can be separated, being a product of factors ?j, one for each particle
state j, not one for each particle. This is the partition function we
used to obtain Eqs. (236) and (237). As was demonstrated there and
earlier, this way of counting states results in the thermodynamics of
a perfect gas. It results also in the Maxwell Bo ltzmann distribution
for the mean number of particles occupying a given particle state j.
This last statement can quickly be shown by obtaining the grand po
tential Q from 5 and then, by partial differentiation by p., obtaining
the mean number of particles in the system,
n MB = a m (? MB ) = w ]2 e( M " e j)AT =  p v
j
N = e ^ kT Z e" e i /kT = L ft,; n r e ( ^# T (245)
where N is equal to PV/kT, thus fixing the value of p=kTln (kT/PV)
£j /kT
In fact ju. acts like a magnitude parameter in the grand
canonical ensemble; its value is adjusted to make  (312/3 jll)tv e Q ua l
to the specified value of N. The quantity f\ j , the mean value of the oc
cupation number for the jth particle state for this ensemble, takes
the place of the particle probabilities (for we can no longer ask what
state a given particle is in). We see that the mean number of particles
in state j, with energy ej, is
 £ i/ kT ) p" £ j/ kT
(246)
— e /kT
which is proportional to the Maxwell Bo ltzmann factor e V
Therefore particles that correspond to this partition function may
be called Maxwell Boltzmann particles (MB particles for short). No
actual system of particles corresponds exactly to this distribution for
all temperatures and densities. But all systems of particles approach
this behavior in the limit of highenough temperatures, whenever the
classical phasespace approximation is valid.
212 STATISTICAL MECHANICS
Before the advent of the quantum theory the volume of phase space
occupied by a single microstate was not known; in fact it seemed rea
sonable to assume that every element of phase space, no matter how
small, represented a separate microstate. If this were the case, the
chance that two particles would occupy the same state was of the sec
ond order in the volume element and could be neglected. Thus for
classical statistical mechanics the procedure of dividing by N! was
valid. Now we know that the magnitude of phasespace volume occu
pied by a microstate is finite, not infinitesimal; it is apparent that
there can be situations in which the system points are packed closely
enough in phase space so that two or more particles are within the
volume that represents a single microstate; in these cases the MB
statistics is not an accurate representation.
Bosons and Fermions
Actual particles are of two types. Both types are indistinguishable
and thus, according to Eq. (2011), have multiplicity factors g v = 1,
rather than (NJ/nJ n 2 ! •••). A state of the system is specified by spec
ifying the values of the occupation numbers nj. Each such state is a
single one; it has no meaning to try to distinguish which particle is in
which state; all we can specify are the numbers nj in each state.
In addition to their indistinguishability, different particles obey dif
ferent rules regarding the maximum value of nj. One set of particles
can pack as many into a given particle state as the distribution will
allow; nj can take on all values from to°°. Such particles are called
bosons; they are said to obey the Bose Einstein statistics (BE for
short). Photons and helium atoms are examples of bosons. For these
particles the g v of Eq. (242) are all unity and the grand partition
function is
?BE= H ex P &AT)2]nj(M6j)
ni,n 2 , ... L j
= y ^(MeJ/kT y e n 2 ( M e 2 )/kT_ = ^ 2
n i n 2
r [le^# T ]" 1 (247)
where we have used Eq. (202) to consolidate the factor sums jj.
Here again the grand partition function separates into factors, one
for each particle state, rather than one for each particle. We note that
the series for the jth factor does not converge unless \± is less than
the corresponding energy ej.
Here again we can calculate the grand potential and the mean num
ber of particles in the system of bosons,
QUANTUM STATISTICS 213
fiBE = kTZ;in[le^ f i )/kT ] = PV
J
* = &; ■j[. (, J' i,/kr l]" 1 (248,
where m is the mean number of particles in the jth particle state.
In this case there is no simple equation fixing the value of /i in terms
of N (or of PV) and T, nor is the relationship between N and Q, = PV
as simple as it was with Eq. (245). Nevertheless, knowing the allowed
energy values €i and the temperature T, we can adjust [i so the sum
of l/e J  lj over all j is equal to N. This value of /i is
then used to compute the other thermodynamic quantities.
Note the difference between the occupation number nj for the boson
and that of Eq. (245) for the MB particle. For higher states, where
ej  \i ^> kT, the two values do not differ much, but for the lower
states, at lower temperatures, where ej  jli is equal to or smaller
than kT, the nj for the boson is appreciably greater than that for the
MB particle (shall we call it a maxwellon?). Bosons tend to "con
dense" into their lower states, at low temperatures, more than do
maxwellons.
The other kind of particle encountered in nature has the idiosyn
crasy of refusing to occupy a state that is already occupied by another
particle. In other words the occupation numbers nj for such particles
can be or 1, but not greater than 1. Particles exhibiting such unso
cial conduct are said to obey the Pauli exclusion principle. They are
called fermions and are said to obey FermiDime statistics (FD for
short). Electrons, protons, and other elementary particles with spin
1/2 are fermions. For these particles g u = 1, but the sum over each
nj omits all terms with n, > 1. Therefore,
3 F D = Zj exp &AT')2j nj(/i ej)
n i> n 2> ••• L J
V»s
*j
[i+^ejVkT] (24 . 9)
Again the individual factors ^ are for each quantum state, rather
than for each particle. The mean values of the occupation numbers
can be obtained from Q as before,
nFD =_ kT ;r ln [ 1 + e (M ej )AT] = _ PV
j
N=^n j; ^[e^^AT+i]" 1 (2410)
j = l
214
STATISTICAL MECHANICS
where again the relation between N and PV (i.e., the equation of
state) is not so simple as it is for maxwellons, and again there is no
simple relationship that determines jul in terms of N; the equation for
N must be inverted to find ju as a function of N.
Comparing the mean number m of particles in state j for fermions
with the hi for MB particles [Eq. (245)], we see that for the higher
states, where €j  \± ^> kT, the two values are roughly equal, but for
the lower states the n^ for fermions is appreciably smaller (for a
given value of jll) than the n, for maxwellons. Fermions tend to stay
away from the lower states more than do maxwellons, and thus much
more than do bosons. In fact, fermions cannot enter a state already
occupied by another fermion; according to the Pauli principle fij can
not be larger than 1.
Comparison among the Three Statistics
The differences between the BE, MB, and FD statistics can be
most simply displayed by comparing the multiplicities gv of the mean
occupation numbers fij. In each case the multiplicities are products
of factors gj(nj), one for each particle state j. The three sets of val
ues are
gj(nj)
1 (nj = or 1)
1 (nj = or 1)
^1 (nj = 0or 1)
(nj = 2,3,...)
= nf < n J > «
(nj=2,3,...)
BE statistics
MB statistics
FD statistics
(2411)
The
g's are identical for
nj = or 1; they differ for the higher values
of the occupation numbers. Bosons have gj = 1 for all values of m;
they don't care how many others are in the same state. Fermions have
gi = for nj > 1; they are completely unsocial. The approximate sta
tistics we call MB has values intermediate between and 1 for nj>l;
these particles are moderately unsocial; the gj tend toward zero as
n^ increases.
In terms of the energy €i of the
of the normalizing parameter
j is
M)AT
1/
e (e 3
 1
jth particle state and the value
the mean number of particles in state
BE statistics
5ri Ve (e i^ T
1/
MB statistics
( ( €j m)At + {\ FD statistics
(2412)
QUANTUM STATISTICS 215
For FD statistics, hj can never be larger than 1; for MB statistics
Rj can be larger than 1 for those states with ji larger than ej; for BE
statistics /x cannot be larger than € x [see discussion of Eq. (247)]
but rij can be much larger than 1 if (e j  ti)/kT is small.
In each case the value of ju. is determined by requiring that the sum
of the rij's, over all values of j, be equal to the mean number N of
particles in the system. If kT is large compared to the energy spac
ings ej + 1  €j, then hjii will not differ much from rij and the sum
for N will consist of a large number of fij's, of slowly diminishing
magnitude. Therefore much of the sum for N will be ' 'carried" by
the nj's for the higher states (j > 1). If, at the same time, N is
small, then all the hj's must be small; even f^ must be less than 1.
For this to be so, (e x  /i) must be larger than kT, so that the terms
( e .  u)/kT
e 3 p ' must all be considerably larger than 1. In this case the
values of the m's, for the three statistics, are nearly equal, and we
might as well use the intermediate MB values, since these provide us
with a simpler set of equations for jul, S, P, C v , etc. [Eqs. (245)] .
In other words, in the limit of high temperature and low density,
both bosons and fermions behave like classical Maxwell Bo ltzmann
particles. For this reason, the fact that classical statistical mechan
ics is only an approximation did not become glaringly apparent until
systems of relatively high density were studied at low temperatures
(except in the case of photons, which are a special case).
When kT is the same size as e 2  e x or smaller, the three statis
tics display markedly different characteristics. For bosons \i be
comes very nearly equal to e 1 (/i = e 1 6, where 5 <^C kT) so that
5i = [>»•">/* 1]" 1
kT/6
and
S r [e (t i ( ' )/kT l] J forj>l
which is considerably smaller than n x if e 2  e x > kT. Therefore at
low temperatures and high densities, most of the bosons are in the
lowest state (j = 1) and
ni  N  V [e (6 3 " €l)/kT  1 ]"'  N, kT 0 (2413)
j = 2
which serves to determine 6, and therefore \i = e x  6. At very low
temperatures bosons "condense" into the ground state. The "conden
sation" is not necessarily one in space, as with the condensation of a
216 STATISTICAL MECHANICS
vapor into a liquid. The ground state may be distributed all over po
sition space but may be "condensed" in momentum space. This will
be illustrated later.
For fermions such a condensation is impossible; no more than one
fermion can occupy a given state. As T — 0, \± must approach e^, so
that rij = Le J K + 1J is practically equal to 1 for j <N (since
e J e* e J N is then very much smaller than 1) and is
much smaller than 1 for j > N (since e J is then very large
compared to 1 for ej >e^). Thus at low temperatures the lowest N
particle states are completely filled with fermions (one per state) and
the states above this "Fermi level" ejsf are devoid of particles.
The behavior of MB particles differs from that of either bosons
or fermions at low temperatures and high densities. The lower states
are populated by more than one particle, in contrast to the fermions,
but they don't condense exclusively and suddenly in just the ground
state, as do bosons. The comparison between the number of particles
per unit energy range, for a gas of bosons, one of fermions and one of
maxwellons, is shown in Fig. 241. We see that fermions pack the
lower N levels solidly but uniformly, that bosons tend to concentrate
in the very lowest state, and that maxwellons are intermediate in dis
tribution. Because of the marked difference in behavior from that of a
classical perfect gas, a gas of bosons or fermions at low temperatures
and high densities is said to be degenerate.
Distribution Functions and Fluctuations
With indistinguishable particles there is no sense in asking the
probability that a specific particle is in state j; all we can ask for is
the probability f j(rij) that nj particles are in state j. These probabil
ities can be obtained from the distribution function of the ensemble,
given in Eq. (234). For
f N^ = fei//3) ex P
^T nj(/i€pAT
j = l
fi( n i) ,f 2 ( n 2 )
*A=tej/»j)e il,( ' X " e ' )AT (2414)
where the factor ?j of the partition function, for the jth particle
state, is given by Eq. (244), (247), or (249), depending upon whether
the particles in the system are MB, BE, or FD particles.
To be specific, the probability that n particles are in state j (we
can leave the subscript off n without producing confusion here) for
the three statistics, is
QUANTUM STATISTICS
217
y = e/kT
1
i
4
/ \ T] =
10
1
>;
^
2
\ 2Vy7^ ^
1 \ \ «• *** """"
\ **^^ l\ ^
\ ^*^ ' X
FD
1
l
Q
X
—»•
—
/ 1 Mi L l~~T
1
1
y = e/kT
Fig. 241. Density of particles per energy range for a gas,
according to the three statistics, for nondegen
erate and degenerate conditions. Dashed curve
2(yA) 1/2 copresponds to a density of one parti
cle per particle state. Area under each curve
equals r\ . See also page 233.
218
STATISTICAL MECHANICS
t, v. r/\ n(/iej)/kT (n+l)(jLL€i
For bosons: fj(n) = e v ^ J"  e ^ J
)/kT
For MB
particles:
fj(n) =
1
^exp
jLL€j
L n k T 
r n( M ej)AT/
For fermions:
fj(») =
o
e CM6j)/kTl
n> 1
(2415)
Reference to Eqs. (2412) shows that the mean value of n is given by
the usual formula,
E rf jw
(2416)
With a bit of algebraic juggling, we can then express the probability
f j(n) in terms of n and of its mean value n* (we can call it n without
confusion here):
fj(n) = <
(5) n /(i+l) n+1
[(H)"/n!] e  "
for bosons
for MB particles
1n if n = 0, = n if n = 1, =0 ifn>l for fermions
(2417)
The distribution function fj(n) for bosons is a. geometric distribution.
The ratio fj(n)/fj(n 1) is a constant, ii/(n + 1); the chance of adding
one more particle to state j is the same, no matter how many bosons
are already in the state. The MB distribution is the familiar Poisson
distribution of Eqs. (1115) and (2310), with ratio fj(n)/fj(n 1) = n/n,
which decreases as n increases. The presence of maxwellons in a
given state discourages the addition of others, to some extent. On the
other hand, fj(n) for FD statistics is zero for n > 1; if a fermion oc
cupies a given state, no other particle can join it (the Pauli principle).
Using these expressions for fj(n) we can calculate the variance
(Anj) 2 of the occupation number m for state j, for each kind of sta
tistics:
(Ani) 2
z>
8,11,0.)
L
n
n 2 fj(n)  (nj) 2
QUANTUM STATISTICS
219
= ■<
r B,(B,+l)
^Hj(l  Sj)
for bosons
for MB particles
for fermions
(2418)
and, from this, obtain the fractional fluctuation Anj/rij of the occupa
tion numbers,
An j/ fi j
]/T+ ( 1 /n j ) for bosons
]/l/nT for MB particles
j/(l/n.j)  1 for fermions
(2419)
The fractional fluctuation is greatest for the least occupied states
(Rj <^C 1). As the mean occupation number increases the fluctuation de
creases, going to zero for fermions as fij — 1 (the degenerate state)
and to zero for maxwellons as
nj *CO
But the standard deviation An
for bosons is never less than the mean occupancy m. We shall see
later that the local fluctuations in intensity of thermal radiation (pho
tons are bosons) are always large, of the order of magnitude of the
intensity itself, as predicted by Eq. (2419).
25
BoseEinstein
Statistics
The previous chapter has indicated that, as the temperature is low
ered or the density is increased, systems of bosons or of fermions
enter a state of degeneracy, wherein their thermodynamic properties
differ considerably from those of the corresponding classical system,
subject to Maxwell Bo ltzmann statistics. These differences are ap
parent even when the systems are perfect gases, where the interaction
between particles is limited to the few collisions needed to bring the
gas to equilibrium. Indeed, in some respects, the differences between
the three statistics are more apparent when the systems are perfect
gases than when they are more complex in structure. Therefore it is
useful to return once again to the system we started to study in Sec
tion 2, this time to analyze in detail the differences caused by differ
ences in statistics. In this chapter we take up the properties of a gas
of bosons. Two different cases will be considered; a gas of photons
(electromagnetic radiation) and a gas of material particles, such as
helium atoms.
General Properties of a Boson Gas
Using Eqs. (248) et seq., we compute the distribution function,
mean occupation numbers, and thermodynamic functions for the gas of
bosons:
f .( n)== [x _ ;(**,)/«] e n ^ e J )/kT 4nj n /(n j + l) n+1 ]
(eiju)AT
n j = £ nfj(n)= [e v V
n =
Q BE = PV = kT In ^ = kT
£
In
(ji6j)/kT
220
BOSEEINSTEIN STATISTICS 221
N = Oft/a/i) TV = E n j = £
3 = 1 j = l
(ejju)AT _ 1
S = 0O/8T) y = (U  N/i  0)/T
OO
U= £ epj; C v = TOS/aT) M y (251)
j=l
where \i must be less than the lowest particle energy e x in order
that the series expansions converge. All these quantities are functions
of the chemical potential /i. For systems in which the mean number
of particles N is specified, the value of \i, as a_f unction of N, V, and
T, is determined implicitly by the equation for N given above. The
value obtained by inverting this equation is then inserted in the other
equations, to give S, U, P, and C v as functions of N, V, and T.
In the case of the photon gas, in equilibrium at temperature T in a
volume V (black body radiation), the number of photons N in volume
V is not arbitrarily specified; it adjusts itself so that the radiation is
in equilibrium with the constant temperature walls of the container.
Since, at constant T and V, the Helmholtz function F = ft + jllN comes
to a minimal value at equilibrium [see the discussion following Eq.
(810)] , if N is to be varied to reach equilibrium at constant T and
V, we must have
(3F/3N) TV = j (ft + jllN) = ju equal to zero (252)
Therefore, for a photon gas at equilibrium, at constant T and V, the
chemical potential of the photons must be zero [see the discussion fol
lowing Eq. (78)] .
Classical Statistics of Black Body Radiation
At this point the disadvantages of a "logical" presentation of the
subject become evident; a historical presentation would bring out more
vividly the way experimental findings forced a revision of classical
statistics. It was the work of Planck, in trying to explain the frequency
distribution of electromagnetic radiation which first exhibited the inad
equacy of the MaxwellBoltzmann statistics and pointed the way to the
development of quantum statistics. A purely logical demonstration,
that quantum statistics does conform with observation, leaves out the
atmosphere of struggle which permeated the early development of
quantum theory, struggle to find a theory that would fit the many new
and unexpected measurements.
222
STATISTICAL MECHANICS
Experimentally, the energy density of blackbody radiation having
frequency between co/27J and (co + dco)/27r was found to fit an empiri
cal formula
de
li
co 3 da)
2 „3
7T C
■fiw/kT
(co 2 kT/77 2 c 3 ) dco
(•Ra) 3 A 2 c 3 )e^ /kT
dco
kT »tict)
kT <Cnu>
where, at the time, "n was an empirical constant, adjusted to fit the
formula to the experimental curves. Classical statistical mechanics
could explain the lowfrequency part of the curve (kT ^>1iio) but could
not explain the highfrequency part (Fig. 251).
1 1 1
i i i
1
/ /
 ; /
/ /
/ /
//
 //
//
i/
y \ \ i
i i
2 4
y = WkT
Fig. 251. The Planck distribution of energy density of
blackbody radiation per frequency range.
Dashed line is Rayleigh Jeans distribution.
Classically, each degree of freedom of the electromagnetic radia
tion should possess a mean energy kT [see the discussion of Eq.
(155)] , so determining the formula for de should simply involve
finding the number of degrees of freedom of the radiation between co
and co + dco. Since the radiation is a collection of standing waves, it
can proceed exactly as was done in Chapter 20, in finding the number
of standing waves in a crystal with frequencies between u)/2u and
(co + dw)/27r [see Eqs. (2010) and (2011)] . In a rectangular enclosure
of sides l x , ly, lz the allowed values of co are
CO
j = 7rc[(k/l x ) 2 + (m/l y ) 2 + ( n /l z ) 2 ]
211/2
(254)
where k, m, n are integers and where c is the velocity of light.
Each different combination of k, m, n corresponds to a different elec
BOSEEINSTEIN STATISTICS 223
tromagnetic wave, a different degree of freedom or, in quantum lan
guage, a different quantum state j for a photon.
By methods completely analogous to those used in Chapter 20, we
find that the number of different degrees of freedom having allowed
values of uh between u) and u> + do; are
dj = (VA 2 c 3 )a> 2 du>, V = l x l y l z (255)
which is twice the value given in Eq. (2011) because light can have
two mutually perpendicular polarizations, so there are two different
standing waves for each set of values of k, m, and n. As mentioned
before, this formula is valid for nonrectangular enclosures of volume
V.
Now, if each degree of freedom carries a mean energy kT, then
the total energy within V, between w and a> + du>, is (kT)dj and the
energy density of radiation with frequency between u>/2tt and
(u> + da>)/277 is
de = (kT/V)dj = (u> 2 kTA 2 c 3 ) du>
which is called the RayleighJeans formula. We see that it fits the
empirical formula (253) at the lowfrequency end (see the dashed
curve of Fig. 251) but not for high frequencies.
As a matter of fact it is evident that the RayleighJeans formula
cannot hold over the whole range of w from to °° , for the integral
of de would then diverge. If this were the correct formula for the en
ergy density then, to reach equilibrium with its surroundings, a con
tainer filled with radiation would have to withdraw an infinite amount
of energy from its surroundings; all the thermal energy in the uni
verse would drain off into highfrequency electromagnetic radiation.
This outcome was dramatized by calling it the ultraviolet catastrophe .
There is no sign of such a fate, so the RayleighJeans formula cannot
be correct for high frequencies. In fact the empirical curve has the
energy density de dropping down exponentially, according to the fac
tor e ' , when "ho; ^> kT, so that the integral of the empirical
expression does not diverge.
Parenthetically, a similar catastrophe cannot arise with waves in a
crystal because a crystal is not a continuous medium; there can only
be as many different standing waves in a crystal as there are atoms
in the crystal; integration over u> only goes to oj m [see Eq. (2613)]
not to °°. In contrast, the electromagnetic field is continuous, not
atomic, so there is no lower limit to wavelength, no upper limit to the
frequency of its standing waves.
A satisfactory exposition (to physicists, at any rate) would be to
proceed from empirical formula (253) to the theoretical model that
224 STATISTICAL MECHANICS
fits it, showing that the experimental findings lead inexorably to the
conclusion that photons obey BoseEinstein statistics. We have not
the space to do this; we shall show instead that assuming photons are
bosons (with \i = 0) leads directly to the empirical formula (253) and
by identifying the empirical constant "ft = h/277 with Planck's constant,
joins the theory of blackbody radiation to all the rest of quantum
theory.
Statistical Mechanics of a Photon Gas
As we have already pointed out in Eq. (252), photons are a rather
special kind of boson; their chemical potential is zero when they are
in thermal equilibrium in volume V at temperature T. Formulas
(251) thus simplify. For example, the mean number of photons in
state j is rij = (e V  l) . But state j has been defined as the
state that has frequency wj/27r, where a>j is given in Eq. (254) in
terms of its quantum numbers. Since a photon of frequency coj/277
has energy ej ="Rcoj, the mean occupation number becomes
n r x/(e^ kT l) (256)
Since there are (V/7r 2 c 3 )ct) 2 do; = dj different photon states (differ
ent standing waves) with frequencies between u>/277 and (u; + du;)/27r,
the mean number of photons in this frequency range in the container
is
^  V a) 2 do; , v
dn  — 5T ~ z — 7T7^ (257)
7i 2 c 3 ■nw/kT 1
The mean energy density de of blackbody radiation in this frequency
range is dn times the energy "fto; per photon, divided by V, which
turns out to be identical with the empirical formula for de given in
Eq. (253). Thus the assumption that photons are bosons with n=
leads directly to agreement with observation.
The frequency distribution of radiation given in Eq. (253) is called
the Planck distribution. The energy density per unit frequency band
increases proportional to co 2 at low frequencies; it has a maximum at
w = 2.82(kT/Ii) [where x = 2.82 is the solution of the equation (3  x)e x
= 3] and it drops exponentially to zero as w increases beyond this
maximum. Measurements have checked all these details; in fact this
was the first way by which the value of h was determined. The mean
number of photons, and the mean energy density, of all frequencies
can be obtained from the following formulas:
Jj^ =2.404; J £^L = £ = 6.494 (258)
BOSE EINSTEIN STATISTICS 225
For example, the mean energy density is
e(T) _ f de _ (kT) 4 ? x 3 dx = K*V
which is the same as Eq. (78), of course, only now we have obtained
an expression for Stefan's constant a in terms of k, h, and c (which
checks with experiment).
The grand potential ft (which also equals F, since (i = 0) is
Q = kT /dn in (l  e^/ kT ) = J" J u > dw ln (i _ e ^/ kT )
= SS/^T= aVT4 =  Ve(T) = F (25  10)
where we have integrated by parts. The other thermodynamic quanti
ties are obtained by differentiation,
...(g) =f,VT, ,gj) i =i.T.= i„T,
(2511)
which also check with Eqs. (78). The mean number of photons of any
frequency in volume V is
 _ r Vk 3 T 3 7 x 2 dx 2.404 /kT\ 3 0.625k 3 3
W J dn 77 2 c 3 n 3 J e*l 7i 2 Icli/ V ~ cH V1
o
(2512)
Statistical Properties of a Photon Gas
We saw earlier that the assumption of classical equipartition of
energy for each degree of freedom of blackbody radiation leads to
the nonsensical conclusion that each container of radiation has an in
finite heat capacity. The assumption that photons are bosons, with
jll = 0, leads to the Planck formula, rather than the Rayleigh Jeans
formula, and leads to the conclusion that the mean energy carried per
degree of freedom of the thermal radiation is
kT ficiM « kT
lUonj = (li Wj /(e tiw / kT  l)  I '
kT
(2513)
226 STATISTICAL MECHANICS
which equals kT, the classical value, only for lowfrequency radiation.
At high frequencies each photon carries so much energy that, even in
thermal equilibrium, very few can be excited (just as with other quan
tized oscillators that have an energy level spacing large compared to
kT), and the mean energy possessed by these degrees of freedom falls
off proportionally to e . Thus, as we have seen, the mean en
ergy e(T) is not infinite as classical statistics had predicted.
As was pointed out at the end of Chapter 24, the fluctuations in a
boson gas are larger than those in a classical gas. For a photon gas
the standard deviation Anj of the number of phot ons in a pa rticular
state j, above and below the mean value fij, is T/ns (nj + 1) and conse
quently the fractional fluctuation is
ha>/2kT
Anj/flj = lAhj + D/hj = e ^/^ A (2514)
This is also equal to the fractional fluctuation of energy density
Aej/ej or of intensity Alj/L of the standing wave having frequency
0)^/277. This quantity is always greater than unity, indicating that the
standard deviation of the intensity of a standing wave is equal to or
greater than its mean intensity.
Such large fluctuations may be unusual for material gases; they
are to be expected for standing waves. If the jth wave is fairly stead
ily excited (i.e., if hj > 1, i.e., if e V < 2) then it will be oscil
lating more or less sinusoidally and its intensity will vary more or
less regularly between zero and twice its mean value, which corre
sponds to a standard deviation roughly equal to its mean value. If, on
the other hand, the standing wave is excited only occasionally, the si
nusoidal oscillation will occur only occasionally and the amplitude will
be zero in between times. In this case the standard deviation will be
larger than the mean. Thus Eq. (2514) is not as anomalous as it might
appear at first.
The variance (Adn) 2 of the number of photons in all the standing
waves in volume V having frequency between a>/27r and (<jj+ du>)/27r is
the sum of the variances of the component states,
rAHnV (a„ yf v " 2 M  Va> 8 dw e" Ba,/kT
(Adn)  (Anj) [^T^j ~ ^^T / WkT _ \»
and the fractional fluctuation of energy in this frequency range is
Adn/dn = VV c 3 /Vu> 2 dw £*>/ 2kT (2515)
The wider the frequency band dcu and the greater the volume consid
BOSEEINSTEIN STATISTICS 227
ered, the smaller is the fractional fluctuation; including a number of
standing waves in the sample ''smooths out" the fluctuations.
Statistical Mechanics of a Boson Gas
When the bosons comprising the gas are material particles, rather
than photons, jll is not zero but is determined by the mean particle
density. The particle energy e is not tiu) but is the kinetic energy
p 2 /2m of the particle if m is its mass. We have already shown [see
Eqs. (192) and (217)] that, for elementary particles in a box of
' 'normal" size, the trans lational levels are spaced closely enough so
that we can integrate over phase space instead of summing over par
ticle states. The number of particle states in an element dx dy dz dp x
x dpy dp z = dVq dVp of phase space is g(dVq dVp/h 3 ) where g is the
multiplicity factor caused by the particle spin. If the spin is s and no
magnetic field is present, g = (2s + 1) different spin orientations have
the same energy e. Therefore the sum for N of Eq. (251) becomes
= (g/h3)/./[e (£ ^ kT l] 1 dV q dV p
= (gV/h 3 ) f d/3 [sin a da ( [e U " M)/kT  1
P 2 dp
(2516)
where angles a and /3 are the spherical angles in momentum space
of Eq. (121).
We can integrate over a and /3 and, since e = (p 2 /2m) or p = ]/2me,
we can change to € for the other integration variable, so
N=2, gV (f) 3/2 J **
(e  M )AT
o e " L
= gV prnpkTj f 1/a ( M /kT) (2517)
* / \ 1 ? z m dz V* / _nx / m + *\ x
f m(*)=— , J e z + x n = L [ e A j^ e >
n n=l
X_»oo
The series for f m converges if x is positive. However we recollect
that with BoseEinstein statistics ji must be less than the lowest en
ergy level, which is zero for gas particles. Therefore /i is negative
and x = ( /i/kT) is positive, and the series does converge.
228 STATISTICAL MECHANICS
It should be pointed out that the change from summation to integra
tion has one defect; it leaves out the ground state e = 0. This term, in
the sum of particle states, is the largest term of all: in the integral
approximation it is completely left out because the density function ^JT
goes to zero there. Ordinarily this does not matter, for there are so
many terms in the sum for N for e small compared to kT (which are
included in the integral) that the omission of this one term makes a
negligible difference in the result. At low temperatures, however, bo
sons "condense" into this lowest state [see Eq. (2413)] and its popu
lation becomes much greater than that for any other state. We shall
find that above a limiting temperature T the ground state is no more
densely populated than many of its neighbors and that it can then be
neglected without damage. Below T , however, the lowest state be
gins to collect more than a normal share of particles, and we have to
add an extra term to the integral for N, corresponding to the number
of particles that have "condensed" into the zeroenergy state.
We should have mentioned this complication when we were discus
sing a photon gas, of course, for the integrals of (259) to (2511) also
have left out the zero energy state. But a photon of zero energy has
zero frequency, so this lowest energy state represents a static elec
tromagnetic field. We do not usually consider a static field as an as
semblage of photons and, furthermore, the exact number of photons
present is not usually of interest; the measurable quantities are the
energy density and intensity. For morematerial bosons, however,
the mean number of particles can be measured directly, so we must
account for the excess of particles in the zero energy state when the
gas is degenerate.
Thermal Properties of a Boson Gas
The value of \i is determined implicitly by the equation
77 s Ni/gV = f 1/2 (x) ; x =  jLt/kT ; 1 1 = h/"/27rmkT
(2518)
which can be inverted to obtain /x as a function of T and 77. When
the parameter 77 is small (low density and/or high temperature), f 1/2
 x u. /kT
has its limiting form e = e and
H kT In (gV/N£) = kT In 7?, 77 —
which is the value for a classical, perfect gas of Eqs. (237). The
computed values of x for larger values of 77 are given in Table 251.
The internal energy U and the grand potential £2 of the gas can also
be expressed as integrals,
BOSEEINSTEIN STATISTICS 229
TABLE 251
Functions for a Boson Gas
V
X
T/T
PV/NkT S/Nk
2C v /3Nk
N c /N
oo
oo
1.000
oo
1.00
0.1
2.342
8.803
0.977
4.784
1.01
1
0.358
1.897
0.818
2.403
1.09
2
0.033
1.195
0.637
1.625
1.19
2.5
0.001
1.030
0.536
1.341
1.26
2.612
1.000
0.513
1.282
1.28
3
0.912
0.447
1.116
1.12
0.129
10
0.409
0.134
0.335
0.33
0.739
30
0.196
0.045
0.112
0.11
0.913
oo
1.000
u =
oo
27rgV(2m/h 2 ) 3 / 2 f
e 3/2
de
=  (NkT/
e (e  /i)AT _ 1
V)^3/2(~^/
ft =
+ PV = 27rkTgV(2m/h 2 ) 3/2
ft T (
f Je de In [l  e W
i  e)/kT~
!«
J
s = (ao/aT) VjLt = (u/t) + (mn/t) = Nk
x +
5 W*)
2 f l/2 (x)
(2519)
where we have integrated the expression for ft by parts to obtain
(2/3)U, and where we have used the equation (d/dx)f m (x) = f m _ ^(x)
to obtain the expression for the entropy S. Compare these formulas
with the ones of Eqs. (2114) for a gas of MB particles. The term x
here corresponds to the logarithmic terms in the formula for S, for
example, and the two expressions for S are identical when f 3/2 = f 1 / 2 ,
i.e., when 77 — 0.
The heat capacity C v could be computed by taking the partial of U
with respect to T, holding N and V constant. But the independent var
iables here are /i, T, and V and, rather than bothering to change var
iables, we use the formula C v = T(3S/3T)^y. The values can be com
puted by numerical differentiation or by series expansion. The impor
tant quantities are tabulated in Table 251 for a range of values of the
parameter r\ . We see that when 77 is small, PV is practically equal
to NkT (the perfect gas law) and C v is practically equal to (3/2)Nk.
Equations (2517) and (2519) show that
230 STATISTICAL MECHANICS
(S/Nk) =  [f 3/ ,( M /kT)/f 1/2 (iu/kT)]  ((i/kT) and
(N/VT 3 / 2 ) = g(277mk/h 2 ) 3/2 f 1/2 (^/kT)
are functions of ju/kT alone. In an adiabatic expansion both N and
S remain constant; therefore in an adiabatic expansion /i/kT and
y T 3/2 sta y cons tant for a boson gas. We can also show that PV 5/3 is
constant during an adiabatic expansion. These results are identical
with Eqs. (412) and (65) for a classical perfect gas. Evidently a bo
son gas behaves like a perfect gas of point particles in regard to adi
abatic compression, although its equation of state is not that of a per
fect gas (Table 251 shows that PV/NkT diminishes as r\ increases).
When 77 is less than unity, a firstorder approximation is
ft  PV = u « NkT[l  (r7/2 5/2 )]
3
NkT
1 
n / h 2 y/ 2 "
2gV WmkT/ .
(2520)
from which we can obtain C v by differentiation of U (since the in
dependent variables are now N, T, and V). The boson gas exhibits a
smaller pressure and a larger specific heat than a classical perfect
gas, at least for moderate temperatures and densities.
The Degenerate Boson Gas
As the density of particles is increased and/or the temperature is
decreased, 77 increases, x = (i/kT decreases and the thermal prop
erties of the gas depart farther and farther from those of a classical
perfect gas, until at 77 = 2.612, jll becomes zero. If 77 becomes larger
than this, Eq. (2517) no longer can be satisfied. For the maximum
value of f!/ 2 (x) is 2.612, for \± = 0, and jll cannot become positive.
The only way the additional particles can be accommodated is to put
them into the hithertoneglected zero energy state mentioned several
pages _back.
If N is held constant and T is reduced, the condensation starts
when 77 = 2.612, and thus when T reaches the value
T = (h 2 /27rmk)(N/2.612gV) 2 / 3 = 3.31(:h 2 /mk)(N/gV) 2/3 (2521)
Any further reduction of T will force some of the particles to con
dense into the zero energy state. In fact the number N x of particles
that can stay in the upper states are those which satisfy Eq. (2517)
with ji = 0.
BOSE EINSTEIN STATISTICS
231
N x = 2.612gV(27rmkT/h 2 ) 3/2 = N(T/T ) 3/2 (2522)
and the rest,
N c = N[l  (T/T ) 3/2 ]
are condensed in the ground state, exerting no pressure and carrying
no energy. Therefore, the thermodynamic functions for the gas in this
partly condensed state are
p V = C2 = u = 0.513NkT(T/T ) 3/2 = 0.086
m 3 / 2 gV
( kT )5/2
or
P = 0.086(m 3/2 g/h 3 )(kT) 5 ^
S = 5U/3T = 1.28Nk(T/T ) 3/2 = c v
(2523)
The pressure is independent of volume, because this is all the pres
sure the uncondensed particles can withstand. Further reduction of
volume simply condenses more particles into the ground state, where
they contribute nothing to the pressure. The heat capacity of the gas
as a function of T has a discontinuity in slope at T , as shown in Fig.
25.2. At high temperatures the gas is similar to an MB gas of point
particles, with C v = (3/2)Nk. As T is diminished C v rises until, at
T = T , it has its largest value, C v = 1.92Nk. For still smaller values
of T, C v decreases rapidly, to become zero at T = 0.
2nR
T/6 v
Fig. 252. Heat capacity of a gas (according to the three
statistics) versus temperature in units of
= (h 2 /mk)(N/gV) 2/3 .
232 STATISTICAL MECHANICS
The ' 'condensed particles are not condensed in position space, as
in change of phase; they are "condensed" in momentum space, at
p = 0, a set of N[l  (T/T ) 3/2 ] stationary particles, distributed at
random throughout the volume V. Liquid helium II acts like a mixture
of a condensed phase (superfluid) plus some ordinary liquid, the frac
tion of superfluid increasing as T is decreased. Since Hell is a liq
uid the theoretical model to account for its idiosyncrasies is much
more complicated than the gas formulas we have developed here. Al
though the theory is not yet complete, the assumption that He 4 atoms
are bosons does explain many of the peculiar properties of Hell.
FermiDirac
Statistics
FermiDirac statistics is appropriate for electrons and other ele
mentary particles that are subject to the Pauli exclusion principle.
The occupation numbers nj can only be zero or unity and the mean
number of particles in state j is the hj of Eq. (2410). In this chap
ter we shall work out the thermal properties of a gas of fermions, to
compare with those of a gas of bosons and with those of a perfect gas
of MB particles, particularly in the region of degeneracy. There are
no FD analogues to photons, with fi= 0.
General Properties of a Fermion Gas
For a gas of fermions, at temperature T in a volume V, the par
ticle energy is € = p 2 /2m as before, and the number of allowed trans 
lational states in the element of phase space dV q dV p is g(dV q dV p /h 3 )
as before (g is the spin multiplicity 2s + 1). Integrating over dVq and
over all directions of the momentum vector, we find the number of
states with kinetic energy between € and e+de is 27rgV(2m/h 2 ) 3/2 vTd€,
as before. Multiplying this by m [Eq. (2410)] gives us the mean num
ber of fermions with kinetic energy between e and € +d€ ,
dN = 277gV(2 m /h 2 )°/2 (e _ ^g, (261)
which is to be compared with the integrand of Eq. (2517) for the bo
son gas and with dN = 27rgV(2m/h 2 ) 3/2 e^ " € ^ kT VT d€ for a perfect
gas of MB particles.
Figure 241 compares plots of dN/d€ for these three statistics for
two different degrees of degeneracy. As r\ = (N/gV)(h 2 /27rmkT) 3/2
varies, the MB distribution changes scale, but not shape. For small
values of 77, the values of ji= xkT for the three cases are all nega
tive and do not differ much in value, nor do the three curves differ
much in shape. In this region the MB approximation is satisfactory.
233
234 STATISTICAL MECHANICS
For large values of 77 the curves differ considerably, and the val
ues of the chemical potential ju differ greatly for the three cases. For
bosons, as we saw in Chapter 25, p is zero and a part of the gas has
''condensed" into the ground state, making no contribution to the en
ergy or pressure of the gas, and being represented on the plot by the
vertical arrow at y = 0. For fermions, ju is positive, and the states
with € less than p. are practically completely filled, whereas those
with € greater than p are nearly empty. Because of the Pauli princi
ple, no more than one particle can occupy a state; at low temperatures
and /or high densities the lowest states are filled, up to the energy
€ = p., and the states for € > p are ne arly empty, as shown by the
curve (which is the parabolic curve 2Vy/?T for y less than x and
which drops to zero shortly thereafter).
The dotted parabola 2Vy/7r corresponds to the level density dN/d€
= 27TgV(2m/h 2 ) 3/2 VT, corresponding to one particle per state. We see
that the curve for MB particles rises above this for 77 large, corre
sponding to the fact that some of the lower levels have more than one
particle apiece. The BE curve climbs still higher at low energies.
The FD curve, however, has to keep below the parabola everywhere.
The conduction electrons in a metal are the most accessible exam
ple of a fermion gas. In spite of the fact that these electrons are mov
ing through a dense lattice of ions, they behave in many respects as
though the lattice were not present. Their energy distribution is more
complicated than the simple curves of Fig. 241 and, because of the
electric forces between them and the lattice ions, the pressure they
exert on the container is much less than that exerted by a true gas;
nevertheless their heat capacity, entropy, and mean energy are re
markably close to the Fermigas values. Measurements on conduction
electrons constitute most of the verifications of the theoretical model
to be described in this chapter.
The Degenerate Fermion Gas
As T approaches zero the FD distribution takes on its fully degen
erate form, with aj.1 states up to the Nth completely filled and all
states beyond the Nth completely empty. In other words, the limiting
value of 11 (call it /i ) is large and positive, and
f 277gV(2m/h 2 ) 3 / 2 V^ d€ < e ^ /i
dN= 1
L0 €>m (262)
where p has the value that allows the integral of dN to equal N,
N = 2;rgV(2m/h 2 ) 3/2 J VT d€ or p = /3(N/V) 2/3 (263)
vwc3680 WABMorse jp 112461
FERMIDIRAC STATISTICS 235
where j3 = (h 2 /2m)(3/47Tg) 2/3 = 5.84 x 10" 38 joulemeter 2 for electrons
(g=2).
Even at absolute zero most of the fermions are in motion, some of
them moving quite rapidly. For an electron gas of density p = Nm/V
kg per m 3 , the kinetic energy of the fastest, pi , is roughly equal to
40p 2/3 electron volts; n /k is approximately equal to 4.5 x 10 5 p 2/3 °K.
In other words the top of the occupied levels (the Fermi level) corre
sponds to the mean energy [= (3/2)kT] of a MB particle in a gas at the
temperature 3 x 10 5 p 2/3 °K. For the conduction electrons in metals,
where p e \ ^ 3 x 10~ 2 kg/m 3 , this corresponds to about 30,000 °K; for
free electrons in a whitedwarf star, where p e \ > 1000, it corresponds
to more than 3 x 10 7 degrees. Until the actual temperature of a Fermi
gas is larger than this value, it remains degenerate. The parameter
7] = 0.752(jLt /kT) 3/2 is a good index of the onset of degeneracy (when
r\ > 1 there is degeneracy).
The internal energy of the completely degenerate gas (which, like
the boson gas, is equal to (3/2)ft at all temperatures), is
jPo « «_ «
U = / € dN = /3N(N/V) 2/3 = f N Mo = fl (264)
P = O/V = f/3(N/V) 5 / 3 ; S o =
Even at absolute zero a fermion gas exerts pressure. If electrons
were neutral particles, their pressure would be about 2.7 x 10 7 p 5/3
atmospheres at absolute zero. Because of the strong electrical attrac
tions to the ions of the crystal lattice, this pressure is largely coun
terbalanced by the forces holding the crystal together.
When T is small compared to /i /k (i.e., when r? is larger than
unity) but is not zero, a firstorder correction to the formulas for
complete degeneracy can be worked out. The results are
U  U + ^(N/MoKkT) 2 = N Mo [ + ±7r 2 (kT/ Mo ) 2 ]
F « U  iTr 2 (N//i )(kT) 2 = I^N^V 2 / 3  ^(kT) 2 NV3y 2 / 3
S  7r 2 Nk(kT/ Mo )  C p * C v  77 2 (k 2 T//3)N 1 / 3 V 2 / 3
P « i3(N/V) 5 / 3 + (7T 2 /6/3)(kT) 2 (N/V) 1/3 (265)
These formulas verify that, as long as T is small compared with
236 STATISTICAL MECHANICS
ju /k, the fermion gas is degenerate, with thermal properties very dif
ferent from those of a classical, perfect gas. The internal energy is
nearly constant, instead of being proportional to T; the pressure is
inversely proportional to the 5/3 power of the volume and its depend
ence on T is small.
The heat capacity of the degenerate gas is proportional to T at low
temperatures, being considerably smaller than the classical value 3Nk
when T is less than /x /k. Thus the C v of the conduction electrons is
small compared to the lattice C v for metals at room temperatures.
However, the heat capacity of the lattice of ions is proportional to T 3
for low temperatures [see Eq. (2016)] so that if T is made small
enough, the linear term, for the conduction electron gas, will predom
inate over the cubic term for the lattice. It is found experimentally
that, below about 3°K the heat capacity of metals is linear in T in
stead of cubic, as are nonconductors. This experimental fact was one
of the first verifications of the theoretical prediction (made by Som
merfeld) that the conduction electrons in a metal behave like a degen
erate Fermi gas.
Behavior at Intermediate Temperatures
When T is considerably larger than /i /k, the FD gas is no longer
degenerate; it has roughly the same properties as the MB gas. For
example, its equation of state at these high temperatures is
PV a NkT[l + (t7/2 5/2 )] « NkT
1 + 3^oAT)^] (266)
which differs from the corresponding result for the boson gas of Eq.
(2520) only by the difference in sign inside the brackets. The pres
sure is somewhat greater than that for a perfect gas; the effect of the
Pauli exclusion principle is similar to that of a repulsive force be
tween the particles.
For intermediate temperatures the thermodynamic properties must
be computed numerically. Referring to Eq. (261), we define a param
eter 77, as with Eq. (2518),
( \ = J_ ( h2 f^ = _1_ ( M 3/2 = _2_ 7 Vu~du
mx) gV \2vmkTJ ~ 3V^T VkT/ JT J u+x +i
= F 1/2 (x) e" X , x00 (267)
where x = (/i/kT) can be considered to be a function of 77 . The other
thermodynamic quantities, being functions of x, are therefore func
tions of 77,
FERMIDIRAC STATISTICS 237
ft = PV = u= NkT(F 3/ ^/77)
3
4 7 u 3 / 2 du
3/W 6 J e u+x +1
x
X » oo
^[f(F 3 ^/F 1/2 ) + x
; f =
NkT[x+(F 3/2 /F 1/2 )]
x=(jLt/kT) (268)
where
F3/2
S = Nk
Values of some of these quantities for a few values of the density
parameter 77 are given in Table 261. The onset of degeneracy cor
responds roughly to 77 = 1. For 77 < 1, PV is practically equal to NkT
and C v nearly equal to (3/2)Nk; the gas is a perfect gas. When 77 > 1,
/i is positive, PV is much larger than NkT, S goes to zero, and C v is
much smaller than (3/2)Nk; the gas is degenerate. The curve for C v
is shown in Fig. 252, in comparison with those for a perfect gas and
a boson gas. These numbers should be compared with those of Table
251, for the boson gas.
TABLE 261
Functions for a Fermion Gas
V
X
kT//i
PV/NkT
S/Nk
2C v /3Nk
00
00
1.000
00
1.000
0.01
4.60
17.81
1.001
7.1
0.997
0.1
2.26
3.841
1.017
4.8
0.989
1
 0.35
0.827
1.174
2.6
0.919
10
 5.46
0.178
2.521
0.85
0.529
100
26.0
0.038
10.48
0.18
0.145
316
56.0
0.008
22.48
0.09
0.084
00
— 00
00
27
Quantum
Statistics for
Complex
Systems
This chapter is a mixed bag. In the first part we discuss the way
one can work out the statistical properties of systems that are more
complex than the simple gases studied in the preceding chapters. Here
we show why helium atoms can behave like elementary BoseEinstein
particles, for example, and why there has to be a symmetry factor a
in Eq. (224).
Wave Functions and Statistics
We cannot go much further in discussing quantum statistics without
talking about wave functions. As any text on quantum mechanics will
state, a wave function is a solution of a Schrodinger equation; its
square is a probability density. For a single particle of mass m in a
potential field <p(r) the equation is H# = €^, where H is the differ
ential operator,
which is applied to the wave function ^(r). The values of the energy
factor e , for which the equation can be solved to obtain a continuous,
single valued, and finite ^>, are the allowed energies €j for the par
ticle. The square of the corresponding solutions ^dr) (the square of
its magnitude, if ^ is complex) is equal to the probability density that
the particle is at the point r. Therefore ^ must be normalized,
fff *j(r) 2 dxdy dz = 1 (272)
The mathematical theory of such equations easily proves that wave
functions for different states i and j are orthogonal,
///*i(r) ^j(r) dx dy dz = unless i = j (273)
238
QUANTUM STATISTICS FOR COMPLEX SYSTEMS 239
The wave function ^j(r) embodies what we can know about the par
ticle in state j. According to quantum theory, we cannot know the par
ticle's exact position; so we cannot expect to obtain a solution of its
classical motion by finding x, y, and z as functions of time. All we
can expect to obtain is the probability that the particle is at r at time
t, which is ^ 2 . The relation between classical and quantum mechan
ics is the relation between the operator H of Eq. (271) and the Ham
iltonian function H(q,p) of Eqs. (139) and (164). For a single parti
cle (the kth one, say)
H ^>e ) = i(pL + eV p kzM r k)
We see that the quantum mechanical operator is formed from the
classical Hamiltonian by substituting (ti/i)(8/8q) for each p. For this
reason we call the H of Eq. (271) a Hamiltonian operator
The generalization to a system of N similar particles is obvious.
If there is no interaction between the particles, the Hamiltonian for
the system is the sum of the singleparticle Hamiltonians,
N
H(p,q) = 2] H k (q ' p)
k=l
and the Schr5dinger equation for the system is
N  2
H* = E*; H=^] H k; H k = ^k +( M r k) (274)
k=l
where
v k 8x 2 + a 2 + az 2
k J k k
The values of E for which there is a continuous, singlevalued, and
finite solution ^ u (r lf r 2 , ... , r^) of Eq. (274) are the allowed values
Ey of the energy of the system. They are, of course, the sums of the
singleparticle energies €\, one for each particle. We have used these
facts in previous chapters [see Eqs. (191) and (231), for example].
A possible solution of Eq. (274) is a simple product of single
particle wave functions,
%(r 19 r 2 , ..., r N ) = *}Jx x ) • *j 2 (r 2 ) .*j N (r N )
N
E„= T €i, (275)
L
k=l
240 STATISTICAL MECHANICS
where j^ stands for the set of quantum numbers of the kth particle.
This would be an appropriate solution for distinguishable particles,
for it has specified the state of each particle; state j x for particle 1,
state j 2 for particle 2, and so on. The square of ^ v is a product of
singleparticle probability densities ^j k ( r k) 2 tnat particle k, which
is in the Jk"th state, is at r^. We should note that for particles with
spin, each \I> has a separate factor which is a function of the spin co
ordinate, a different function for each different spin state. Thus coor
dinate r^ represents not only the position of the particle but also its
spin coordinate, and the quantum numbers represented by j^ include
the spin quantum number for the particle.
Symmetric Wave Functions
This product wave function, however, will not do for indistinguish
able particles. What is needed for them is a probability density that
will have the same value if particle 1 is placed at point r (including
spin) as it has if particle k is placed there. To be more precise, we
wish a probability density ^(i^, r 2 , ... , tn) 2 which is unchanged in
value when we interchange the positions (and spins) of particle 1 and
particle k, or any other pair of particles. The simple product wave
function of Eq. (275) does not provide this; if ] x differs from j^, then
interchanging r x and r^ produces a different function.
However, other solutions of Eq. (274), having the same value E^ of
the energy of the system as does solution (275), can be obtained by
interchanging quantum numbers and particles. For example,
^(r N ,r N _!, ...,r 1 ) = ^ JN (r 1 )^ JN _ 1 (r 2 ) ... ^(i^)
is another solution with energy E u . There are N! possible permuta
tions of N different quantum numbers among N different particle wave
functions. If several different particles have the same quantum num
bers, if nj particles are in state j, for example, then there are
(Nl/n^r^! ...) [compare with Eq. (219)] different wave functions ^ v
which can be obtained from (275) by permuting quantum numbers and
particles.
Therefore a possible solution of Eq. (274), for the allowed energy
E^, would be a sum of all the different product functions that can be
formed by permuting states j among particles k. Use of Eqs. (272)
and (273) can show that for such a sum to be normalized, it must be
multiplied by Vn x \n 2 I/N! . However, such details need not disturb
us here; what is important is that this sum is a solution of Eq. (274)
for the system state v with energy E p , which is unchanged in value
when any pair of particle coordinates is interchanged (the change re
arranges the order of functions in the sum but does not introduce new
QUANTUM STATISTICS FOR COMPLEX SYSTEMS 241
terms). Therefore, its square is unchanged by such interchange and
the wave function is an appropriate one for indistinguishable particles.
For such a wave function it is no longer possible to talk about the
state of a particle; all particles participate in all states; all we can
say is that nj particles are in state j at any time. Such a wave func
tion is said to be symmetric to interchange of particle coordinates.
A few examples of symmetric wave functions for two particles are
^(rj^dg and (l/j/5)fei(ri)¥2(*a) + ^iW^fri)]; a few for three par
ticles are ^(r^dgtf^rg) or (l/f/3) [^(r^foJtf.Orj) +
^(rJ^W^dg + ^rg^W^W]; and so on
Antisymmetric Wave Functions
However, since our basic requirement is that of symmetry for the
square of ^, we have an alternative choice, that of picking a wave
function antisymmetric with respect to interchange of particle coor
dinates, which changes its sign but not its magnitude when the coordi
nates of any pair are interchanged. The square of such a ^ also is
unchanged by the interchange. Such an antisymmetric solution can be
formed out of the product solutions of Eq. (275), but only if all parti
cle ^'s are for different states. If every jk differs from every other
j, then an antisymmetric solution of Eq. (274), with energy E u , is
the determinant
*,
VN!
*j 2 W *j a <r.)
*j N (r * J
*j 2 (r N )
*j N (r 2 )
*j N < r N)
(276)
The properties of determinants are such that an interchange of any
two columns (interchange of particle coordinates) or of any two rows
(interc hang e of quantum numbers) changes the sign of ^ p . The proof
that 1]/nT must be used to normalize this function is immaterial here.
What is important is that another whole set of wave functions, satis
fying the requirements of particle indistinguishability, is the set of
functions that are antisymmetric to interchange of particle coordinates,
For this set, no state can be used more than once (a determinant with
two rows identical is zero).
By now it should be apparent that the two types of wave functions
correspond to the two types of quantum statistics. Wave functions for
a system of bosons are symmetric to interchange of particle coordi
242 STATISTICAL MECHANICS
nates; any number of particles can occupy a given particle state. Wave
functions for fermions are antisymmetric to interchange of particle
coordinates; because of the antisymmetry, no two particles can occupy
the same particle state (which is the Pauli exclusion principle). Both
sets of wave functions satisfy the indistinguishability requirement—
that the square of ^ be symmetric to interchange of particle coordi
nates. A simple application of quantum theory will prove that no sys
tem of forces which are the same for all particles will change a sym
metric wave function into an antisymmetric one, or vice versa. Once
a boson, always a boson, and likewise for fermions. It is an interesting
indication of the way that all parts of quantum theory "hang together"
that the fact that the quantity of physical importance is the square of
the wave function should not only allow, but indeed demand, two dif
ferent kinds of statistics, one for symmetric wave functions, the other
for antisymmetric.
We have introduced the subject of symmetry of wave functions by
talking about a system with no interactions between particles. But this
restriction is not binding. Suppose we have a system of N particles,
having a classical Hamiltonian H(q,p) which includes potential ener
gies of interaction 0(r k i), which is symmetric to interchange of parti
cles (as it must be if the particles are identical in behavior). Corre
sponding to this H is a Hamiltonian operator H, which includes the
interaction terms 0(r k i) and in which each p k is changed to (ft/i)
x (3/8q k ); this operator is also symmetric to interchange of particle
coordinates. There are then two separate sets of solutions of the gen
eral Schrodinger equation
H*( ri ,r 2 , ...,r N ) = E*
One set is symmetric to interchange of coordinates,
*(r lt ... , rj, ... , r£, ... , r N ) = ^(r^ ... , r*, ... , rj, ... , r N )
and the other is antisymmetric,
*(ri, ••• , r k , ... , ri, ... , r N ) =  *(r 1? ... , rj, ... , r k , ... , r N )
When there are particle interactions, the two sets usually have differ
ent allowed energies E^; in any case a function of one set cannot
change into one of the other set.
The symmetric set represents a system of bosons and the antisym
metric set represents a system of fermions. It is by following through
the requirements of symmetry of the wave functions that we can work
out the idiosyncrasies of systems at low temperatures and high densi
ties. By this means we can work out the thermal properties of systems
with strong interactions, mentioned early in Chapter 24. At high tern
QUANTUM STATISTICS FOR COMPLEX SYSTEMS 243
peratures and/or low densities, Maxwell Boltzmann statistics is ade
quate and we need not concern ourselves as to whether the wave func
tion is symmetric or antisymmetric.
Wave Functions and Equilibrium States
If a system is made up of N particles of one kind and M of another,
the combined wave function of the whole is a product of an appro
priately symmetrized function of the N particles, times another for
the M particles, with its proper symmetry. Thus, for a gas of N hy
drogen atoms, the complete wave function would be an antisymmetric
function of all the electronic coordinates (including spin) times another
antisymmetric for all the protons (including their spins); for both elec
trons and protons are fermions.
This completely antisymmetrized wave function is appropriate for
the state of final equilibrium, when we have waited long enough so that
electrons have exchanged positions with other electrons, from one
atom to the other, so that any electron is as likely to be around one
proton as another. However, it takes a long time for electrons to in
terchange protons in a rarefied gas. Measurements are usually made
in a shorter time, and correspond to an intermediate "metastable"
equilibrium, in which atoms as a whole change places, but electrons
do not interchange protons. A wave function for such a metastable
equilibrium is one made up of separate hydrogen atomic wave func
tions, symmetrized with regard to interchanging atoms as individual
subsystems.
Thus it is usually more appropriate to express the wave function
for a system of molecules in terms of each molecule as a separate
subsystem, taking into account the effects of exchanging molecule with
molecule as individual units, and arranging it as though the individual
particles in one molecule never interchange with those in another.
The total wave function is thus put together out of products of molecu
lar wave functions, according to the appropriate symmetry for inter
change of whole molecules, each molecular wave function organized
in regard to the appropriate symmetry for exchange of like particles
within the molecule.
Actually the differentiation between " metastable*' and "longterm"
equilibrium in a gas is academic in all but a few cases. By the time
the temperature has risen beyond the boiling point of most gases, so
that the system is a gas, both the translational and rotational motions
of the molecules can be treated classically, as was shown in Chapters
21 and 22, and questions of symmetry no longer play a role. Only a
few cases exist where the relation between wavefunction symmetry
and statistics is apparent. These cases each involved puzzling dis
crepancies with the familiar classical statistics; their resolution con
stituted one of the major vindications of the new statistics.
244 STATISTICAL MECHANICS
One such case is for helium, gas and liquid. Normal helium (He 4 )
is made up of a nucleus of two protons and two neutrons, surrounded
by two electrons. In the bound nuclear state the spins of each heavy
particle pair are opposed, so that the net spin of the He 4 nucleus is
zero, as is the net spin of the electrons in the lowest electronic state.
Since protons, neutrons, and electrons are each fermions, the com
bined wave function for, say, two helium atoms could be a product of
three antisymmetric functions, one for all four electrons, another for
four protons, and a third for the four neutrons. This wave function
would correspond to "longterm" equilibrium, since it includes the
possibility of interchange of neutrons, for example, between the two
nuclei. A more realistic wave function would be formed of products
of separate atomic wave functions.
For example, we could assume that electrons 1 and 2 were in atom
a, electrons 3 and 4 in atom b, and similarly with the neutrons and
protons. The electronic wave function for atom a would then be anti
symmetric for interchange of electrons 1 and 2, that for atom b anti
symmetric for interchange of 3 and 4. We would not consider inter
changing 1 and 3 or 1 and 4 separately, only interchanging atom a as
a whole with atom b. Since interchanging atoms interchanges two elec
trons, and two protons and two neutrons, the effect of the interchange
would be symmetric, since (1) 2 = +1. Therefore the system wave
function should be symmetric for interchange of atoms; He 4 atoms
should behave like bosons. As we mentioned at the end of Chapter 25,
liquid He 4 does exhibit "condensation" effects of the sort predicted
for BE particles at low temperatures. In contrast He 3 , which has only
one neutron in its nucleus instead of two, and thus should not behave
like a boson, does not exhibit condensation effects.
Other molecules also should behave like bosons (those with an even
number of elementary particles) but they become solids before they
have a chance to exhibit the effects of degeneracy. Normal helium (He 4 )
is the only substance with small enough interatomic forces to be still
fluid at temperatures low enough for boson condensation to take place;
and once started, the condensation prevents solidification down to (pre
sumably) absolute zero. At pressures above 25 atm He 4 does solidify,
and none of the condensation effects are noticeable.
Electrons in a Metal
Another system in which the effects of quantum statistics are notice
able is the metallic crystal. In the case of nonmetallic crystals the
usual assumption is valid— that each atom acts as a unit and that there
is not sufficient time (during most experiments) for the atoms to
change places. Consequently the Debye theory of crystal heat capaci
ties does not consider the consequences of indistinguishability of the
constituent atoms. In the case of magnetic effects, however, where
QUANTUM STATISTICS FOR COMPLEX SYSTEMS 245
the spins of some electrons are more strongly coupled to their coun
terparts on other atoms than to their neighbors in the same atom, the
symmetry of the spin parts of these wave functions must be consid
ered.
In the case of metals, the more tightly bound electrons, in the inner
shells of the lattice ions, do not readily move from ion to ion. But the
outer electrons can change places with their neighbors easily. Conse
quently the complete wave function for the metallic crystal can be
written as a product of individual ionic wave functions, one for each
ion in the lattice (with appropriate symmetry within each ionic factor)
times an antisymmetric function for all the conduction electrons. The
individual electron wave functions ^i(r^) are not very similar to the
standing waves of free particles [see Eq. (215)] ; after all, they are
traveling through a crystal lattice, not in forcefree space. But they
can exchange with their neighbors rapidly enough so that it is not pos
sible to specify which electron is on which ion.
The combined wave function for the conduction electrons thus must
be an antisymmetric combination like that of Eq. (276), which means
that the usual type of FD degeneracy will take place over about the
same range of temperatures (0 to about 1000°) as if these electrons
were free particles in a gas. Since the thermal properties of a degen
erate Fermi gas depend more on the degeneracy than on the exact
form of the wave functions, these conduction electrons behave more
like a pure Fermi gas than anyone expected (until it was worked out
by Sommerfeld). In their electrical properties, of course, the effects
of their interaction with the lattice ions becomes important; but even
here the effects of degeneracy are still controlling.
Ortho and Parahydrogen
As mentioned a few pages ago, a system composed of hydrogen
molecules (H 2 ) behaves like an MB gas as far as its trans lational en
ergy goes. Each molecule behaves as though it is an indivisible sub
system and the whole wave function can be considered to be a product
of single molecule wave functions, each of them being in turn products
of trans lational, rotational, vibrational, and electronic wave functions,
corresponding to the separation of energies of Eq. (211). Each of
these molecular factors of course must have the symmetry or antisym
metry required by the particles composing the molecule. For exam
ple, because protons are fermions, interchange of the two protons
composing the molecule must change the sign of the wave function.
(This is not the case with the HD molecule, where the two nuclei are
dissimilar.)
Each proton has a position, relative to the center of mass of the
molecule, and a spin. If the two spins are opposed, so that the total
nuclear spin of the molecule is zero (singlet state) the spin part of the
246 STATISTICAL MECHANICS
nuclear wave function is antisymmetric. Therefore the space part of
the nuclear wave function must be symmetric, in order that the prod
uct of both will change sign when the two are interchanged. On the
other hand if the spins are parallel, so that the total nuclear spin is 1
(triplet state) the spin factor is symmetric and the space part of the
nuclear wave function must be antisymmetric. Now the factor in the
molecular wave function which achieves the interchange of position of
the nuclei is the rotational factor, which is a spherical harmonic of
the angles denoting the direction of the axis through the nuclei. Inter
changing the positions of the nuclei corresponds to rotating the axis
through 180°.
As mentioned in connection with Eq. (221) the allowed values of
the square of the angular momentum of the molecule are 1i 2 £U + 1),
where I is the order of the spherical harmonic in the corresponding
rotational wave function. It turns out that those spherical harmonics
with even values of i are symmetric with respect to reversing the
direction of the molecular axis; those with odd values of £ change sign
when the axis is rotated 180°. The upshot of all this is that those H 2
molecules which are in the singlet nuclear state, with opposed spins,
can only have even values of I and those in the triplet nuclear state
can have only odd values of i.
The nuclear spins are well protected from outside influences, and
it is an exceedingly rare collision which disturbs them enough for the
spin of one molecule to affect that of another. Therefore, the molecule
that is in the singlet nuclear state stays for days in the singlet state
and is practically a different molecule from one in a triplet nuclear
state. Only after a long time (or in the presence of special catalysts)
do the spins exchange from molecule to molecule, so that the whole
system finally comes to overall, longterm equilibrium.
The two kinds of H 2 are permanent enough to be given different
names; the singlet type, with even values of £, is called parahydrogen.
Its rotational partition function is
Z rot = ?. (2£ + l)exp[6» rot jeU + l)/T] (277)
£=0,2,4, ..
instead of the sum over all values of 1 as in Eq. (221), which is valid
for nonsymmetrical molecules. The triplet kind is called orthohydro
gen. Its partition function is
z °t = £ (21 +l)exp[0 rot 1(1 + 1)/T] (278)
1 = 1,3,5,...
Since the multiplicity of the singlet state (parahydrogen) is 1 and that
of the triplet (ortho hydrogen) is 3, hydrogen gas behaves as though it
were a mixture of one part, (1/4)N, of parahydrogen to three parts,
(3/4)N, of ortho hydrogen.
QUANTUM STATISTICS FOR COMPLEX SYSTEMS 247
When the heat capacity of this mixture is measured in the usual
way, the two kinds of hydrogen act like separate substances and the
heat capacity C v = T(3 2 F/8T 2 ) V is a sum of two terms,
,rot
NkT^ ;;~r Ill ?:■" , + t " : ;':„, in v;
4 3T 2
rot 4 ST 2
)
rot/
and not the single term
C rot = NkT(^ln Z t )
v \3T 2 rot/
(279)
(2710)
which was used in Chapter 22, was plotted in Fig. 221, and is valid
for molecules with nonidentical nuclei. There is no difference between
the two formulas in the classical limit of T > ro t> where each par
tial is equal to unity; the result is NkT in both cases. However, at
low temperatures there is a difference between (279) and (2710),
which is plotted in Fig. 271 [the dotted curve is for Eq. (2710), the
solid one for (279), which is the one predicted for H 2 )] . The circles
100° T 200° 300°
Fig. 271. The rotational part of the heat capacity of hy
drogen gas (H 2 ). It differs from the dashed curve
(identical with Fig. 221) because the two nuclei
are indistinguishable protons.
are the measured values, which definitely check with the assumption
that para and orthohydrogen act like separate substances.
If the heat capacities are measured very slowly, in the presence of
a catalyst to speed the exchange of nuclear spin, then the two varieties
of H 2 do not behave like separate substances and the appropriate par
tition function is
248 STATISTICAL MECHANICS
Zrot =  Tj (2A+l)exp[e rot l(l + l)/T]
I even
+  J2 (2l+l)exp[6 rot i(l + l)/T] (2711)
j^odd
and the heat capacity is obtained by inserting this in Eq. (2710). This
results in a different curve, which is not obtained experimentally with
out great difficulty. The metastable equilibrium, in which the two
kinds of hydrogen behave as though they were different substances, is
the usual experimental situation.
The nonsymmetric HD molecule, of course, has no ortho or para
varieties; the two nuclei are not identical. The curve of Fig. 221, or
the dotted curve of Fig. 271 (with a different ro t of course) is the
appropriate one here, and the one that corresponds to the measure
ments.
On the other hand the D 2 molecule again has identical nuclei. The
deuterium nucleus has one proton and one neutron and a spin of 1. Ex
changing nuclei exchanges pairs of fermions, thus the wave function
should be symmetric to exchange of nuclei. The molecule with anti
symmetric spin factor (paradeuterium) is also antisymmetric in re
gard to spatial exchange (i = 1,3,5, ...) and the one with symmetric
spin factor (orthodeuterium) is also symmetric in the spherical har
monic factor (i= 0,2,4, ...). There are twice as many ortho states as
para states. Thus one can compute a still different curve for the C v
for D 2 gas, which again checks with experiment. Because of the deep
lying requirements of symmetry of the wave function, the spin orien
tations of the inner nuclei, which can only be measured or influenced
with the greatest of difficulty, reach out to affect the gross thermal
behavior of the gas. These symmetry effects also correctly predict
the factors a in Eq. (224) for polyatomic molecules.
Thus the effects of quantum statistics turn up in odd corners of the
field, at low temperatures and for substances a part of which can stay
gaslike to lowenough temperatures for the effects of degeneracy to
become evident. For the great majority of substances and over the
majority of the range of temperature and density, classical statistical
mechanics is valid, and the calculations using the canonical ensemble
of Chapters 19 through 22 quite accurately portray the observed re
sults. The situations where quantum statistics must be used to achieve
concordance with experiment are in the minority (luckily; otherwise
our computational difficulties would be much greater). But, when they
are all considered, these exceptional situations add up to exhibit an
impressive demonstration of the fundamental correctness of quantum
statistics.
References
The texts listed below have been found useful to the writer of this
volume. They represent alternative approaches to various subjects
treated here, or more complete discussions of the material.
E, Fermi, "Thermodynamics, " Prentice Hall, New York, 1937, is
a short, readable discussion of the basic concepts.
W. P. Allis and M. A. Herlin, "Thermodynamics and Statistical
Mechanics," McGrawHill, New York, 1952, presents some alterna
tive approaches.
F. W. Sears, "Introduction to Thermodynamics, the Kinetic Theory
of Gases, and Statistical Mechanics, " Addison Wesley, Reading, Mass.,
1953, also provides some other points of view.
H. B. Callen, "Thermodynamics, " Wiley, New York, 1960, is a
"postulational" development of the subject.
Charles Kittel, "Elementary Statistical Physics," Wiley, New York,
1958, contains short dissertations on a number of aspects of thermo
dynamics and statistical mechanics.
J. C. Slater, "Introduction to Chemical Physics," McGrawHill,
New York, 1939, has a more complete treatment of the application of
statistical mechanics to physical chemistry.
L. D. Landau and E. M. Lifchitz, "Statistical Physics," Addison
Wesley, Reading, Mass., 1958, includes a thorough discussion of the
quantum aspects of statistical mechanics.
249
Problems
1. Suppose you are given a tank of gas that obeys the equation of
state (34), a calibrated container that varies (slightly) in volume, in
an unknown way, with temperature, and an accurate method of meas
uring pressure at any temperature. How would you devise a thermom
eter that measures the perfect gas scale of temperature ? Could you
also determine the constants a and b in the equation of state of the
gas?
2. The coefficient of thermal expansion (3 and the compressibility
k of a substance are defined in terms of partial derivatives
is  (i/v)(av/8T) p k = (i/v)(av/ap) T
(a) Show that (8/3/3 P )t = ~{dh/dT)p and that (j3/k) = (3P/8T) V for
any substance.
(b) It is found experimentally that, for a given gas,
RV 2 (Vnb) V 2 (V  nb) 2
P RTV 3  2an(V  nb) 2 K nRTV 3  2an 2 (V  nb) 2
where a and b are constants, and also that the gas behaves like a
perfect gas for large values of T and V. Find the equation of state
of the gas.
3. A gas obeys equation of state (34). Show that for just one crit
ical state, specified by the values T c and V c , both (dP/dV)^ and
(8 2 P/3V 2 )t are zero. Write the equation of state giving P/P c in
terms of T/T c and V/V c . Plot three curves for P/P c as a function
of V/V c , one for T = (1/2)T C , one for T = T c , and one for T = 2T C .
What happens physically when the equation indicates three allowed val
ues of V for a single P and T?
4. Suppose that all the atoms in a gas are moving with the same
speed v, but that their directions of motion are at random.
(a) Average over directions of incidence to compute the mean num
250
PROBLEMS
251
ber of atoms striking an element of wall area dA per second (in
terms of N, V, v, and dA) and the mean momentum per second im
parted to dA.
(b) Suppose, instead, that the number of atoms having speeds be
tween v and v + dv is 2N[1  (v/v m )] (dv/v m ) for v<v m (the direc
tions still at random). Calculate for this case the mean number per
second striking dA and the mean momentum imparted per second, in
terms of N, V, v m , and dA. Show that Eq. (24) holds for both of
these cases.
5. A gas with Van der Waals' equation of state (34) has an inter
nal energy
U
nRT  (an 2 /V) + U (
Compute Cy and Cp as functions of V and T and compute T as a
function of V for an adiabatic expansion.
6. An ideal gas for which Cy = (5/2)nR is taken from point a to
point b in Fig. P6, along three paths, acb, adb, and ab, where P 2
= 2P X , V 2  2Vi.
J
c X
b
^w
d
^T 2
a
^ >
^
.T,
Fig. P6
(a) Compute the heat supplied to the gas, in terms of N, R, and T l9
in each of the three processes.
(b) What is the heat capacity of the gas, in terms of R, for the
process ab?
7. A paramagnetic solid, obeying Eqs. (36), (38), and (48) and
having a heat capacity C v = nAT 3 , is magnetized isothermally (at con
stant volume) at temperature T from am = to a maximum magnetic
field of 3C m . How much heat must be lost? It is then demagnetized
adiabatically (at constant volume) to 9TC
temperature T x of the solid, in terms of JC m , T , A, and D
you explain away the fact that, if JC m is large enough or T small
again. Compute the final
How do
252
THERMAL PHYSICS
enough, the formula you have obtained predicts that T x should be neg
ative ?
8. Derive equations for (3V/3T)p and (3V/8P)x analogous to Eqs.
(44) and (46). Obtain an expression for (8H/8P)x For a perfect gas,
with Cp = (5/2)nRT and (Cp  Cy)/(aV/8T)p = P, integrate the par
tials of H to obtain the enthalpy.
9. Figure P9 shows a thermally isolated cylinder, divided into
two parts by a thermally insulating, frictionless piston. Each side
contains n moles of a perfect gas of point atoms. Initially both sides
have the same temperature; heat is then supplied slowly and reversi
bly to the left side until its pressure has increased to (243P /32).
(a) How much work was done on the gas on the right ?
(b) What is the final temperature on the right ?
(c) What is the final temperature on the left ?
(d) How much heat was supplied to the gas on the left ?
10. The ratio Cp/Cy for air is equal to 1.4. Air is compressed
from room temperature and pressure, in a diesel engine compression,
to 1/15 of its original volume. Assuming the compression is adiabatic,
what is the final temperature of the gas ?
11. Compute AQ 12 and aQ 43 , for a Carnot's cycle using a perfect
gas of point particles, in terms of nR and T^ and T c . Using the per
fectgas scale of temperature, show that AW 23 = AW 41 . Show that the
efficiency of the cycle is (Th  T c )/Th and thus prove that the perfect
gas temperature scale coincides with the thermodynamic scale of Eq.
(55).
12. A magnetic material, satisfying Eqs. (48) and (38) has a con
stant heat capacity, Cy^ = C. It is carried around a Carnot cycle
shown in Fig. P12, gn being reduced isothermally from 9fR 1 to 3TC 2 at
T^, then reduced adiabatically from 9Ti 2 to 9TC 3 , when it has tempera
ture T c , then remagnetized isothermally at T c to 3Ti 4 , and thence
adiabatically back to T^ and am 1 .
(a) Express 2HX 3 in terms of T^, T c , D, C, and 9TC. 2 and relate 2TC 4
similarly with m x .
(b) How much heat is absorbed in process 12? How much given
off in process 34?
(c) How much magnetic energy dW is given up by the material in
each of the four processes ? Show that dW 23 = dW 41 .
PROBLEMS
253
(d) Show that the efficiency of the cycle heat magnetic energy is
(T h  T c )/T h .
m
Fig. P12
13. When a mole of liquid is evaporated at constant temperature T
and vapor pressure P V (T), the heat absorbed in the process is called
the latent heat L v of vaporization. A Carnot cycle is run as shown in
Fig. P13, going isothermally from 1 to 2, evaporating n moles of
liquid and changing volume from V x to V 2 , then cooling adiabatically
y by evaporating an additional small amount of
T  dT, from V 3 to V 4 , and
dT, P V  dP
to T
liquid, then recondensing the n moles at
thence adiabatically to Py, T again.
P
"P
1 T 2
> v dp v
4 T  dT 3
1 1
Fig. P13
(a) Show that V s
Vi = V g
V^ where Vg is the volume occupied
by n moles of the vapor and V£ the volume of n moles of the liquid
and that if dT is small enough, V 3  V 4 * V 2  V x .
(b) Find the efficiency of the cycle, in terms of dPy, Vg  V^,
and nLy.
(c) If this cycle is to have the same efficiency as any Carnot cycle,
this efficiency must be equal to (T h  T c )/Th = dT/T. Equating the
two expressions for efficiency, obtain an equation for the rate of
change dPy/dT of the vapor pressure with temperature in terms of
Vg  Vg, n, Ly, and T.
14. Work out the Carnot cycle with a gas of photons, obeying Eqs.
(78).
254
THERMAL PHYSICS
15. An ideal gas, satisfying Eqs. (47) and (412) is carried around
the cycle shown in Fig. P15; 12 is at constant volume, 23 is adia
batic, 31 is at constant pressure, V 3 is 8V lf and n moles of the gas
are used.
5
2
 1
k ^^^
1
1
v — >
Fig. P15
(a) What is the heat input, the heat output, and the efficiency of the
cycle, in terms of P 1? V\, n, and R?
(b) Compare this efficiency with the efficiency of a Carnot cycle
operating between the same extremes of temperature.
16. An amount of perfect gas of one kind is in the volume CjV of
Fig. P16 at temperature T and pressure P, separated by an im
pervious diaphragm D from a perfect gas of another kind, in volume
C 2 V and at the same pressure and temperature (C x + C 2 = 1). The vol
ume V is isolated thermally. What is the entropy of the combination?
Ci + C 2 = l
Fig. P16
Diaphragm D is then ruptured and the two gases mix spontaneously,
ending at temperature T, partial pressure CjP of the first gas, C 2 P
of the second gas, all in volume V. What is the entropy now? Devise
a pair of processes, using semipermeable membranes (one of which
will pass gas 1 but not 2, the other which will pass 2 but not 1), which
will take the system from the initial to the final state reversibly and
thus verify the change in entropy. What is the situation if gas 1 is the
same as gas 2?
17. Two identical solids, each of heat capacity Cy (independent of
T), one at temperature T + t, the other at temperature T  1, may
PROBLEMS 255
be brought to common temperature T by two different processes:
(a) The two bodies are placed in thermal contact, insulated ther
mally from the rest of the universe and allowed to reach T sponta
neously. What is the change of entropy of the bodies and of the uni
verse caused by this process?
(b) First a reversible heat engine, with infinitesimal cycles, is
operated between the two bodies, extracting work and eventually bring
ing the two to comm on temp erature. Show that this common tempera
ture is not T, but VT 2  t 2 . What is the work produced and what is
the entropy change in this part of process b? Then heat is added re
versibly to bring the temperature of the two bodies to T. What is the
entropy change of the bodies during the whole of reversible process b?
What is the change in entropy of the universe ?
18. Show that T dS = C v dT + T(aP/3T) V dV, and T dS =
C V (8T/8P) V dP + C p 0T/3V)p dV.
19. A paramagnetic solid, obeying Eqs. (36), (38), and (48), has
a heat capacity C p q(T) (at zero magnetic field) dependent solely on
temperature. First, show that
T dS = C p dT  T(8V/3T)p dP + T(d3tl/dT) x d3C
and, analogous to Eq. (813), that (aC P5c /aac) T p = T(3 2 9fTl/8T 2 ) ac . From
this, show that
S = f (Cp()/ T ) dT  nD(3C/T) 2  p V P
and thence obtain G and U as functions of T, P, and 5C. Obtain S as
a function of T, V, and arc and thence obtain F and U as functions of
T, V, and 971.
20. A gas obeys the Van der Waals' equation of state (34) and has
heat capacity at constant volume Cy = (3/2)nR. Write the equation of
state in terms of the quantities t= T/T c , p = P/P c , and v = V/V c ,
where T c = 8a/27Rb, P c = a/27b 2 , V c = 3nb (see Problem '3). Calcu
late T C S/P C V C in terms of t and v, likewise F/P C V C and G/P C V C .
For t = 1/2 plot p as a function of v from the equation of state. Then,
for the same value of t, calculate and plot G/P C V C as a function of p,
by graphically finding v for each value of p from the plot, and then
computing G/P C V C for this value of v (remember that for some val
ues of p there are three allowed values of v). The curve for G/P C V C
crosses itself. What is the physical significance of this?
21. Ten kilograms of water at a temperature of 20 °C is converted
to superheated steam at 250° and at the constant pressure of 1 atm.
Compute the change of entropy of the water. Cp (water) = 4180 joules/
kgdeg. L v (at 100°C) = 22.6 x 10 5 joules/kg. Cp (steam) = 1670 +
0.494T + 186 x 10 6 /T 2 joules/kgdeg.
256 THERMAL PHYSICS
22. Assume that near the triple point the latent heats L m and L v
are independent of P, that the vapor has the equation of state of a per
fect gas, that the volume of a mole of solid or liquid is negligible com
pared to its vapor volume, and that the difference V4  V s is positive,
independent of P or T and is small compared to nL m /T. Using these
assumptions, integrate the three ClausiusClapeyron equations for the
vaporpressure, sublimationpressure and meltingpoint curves.
Sketch the form of these curves on the PV plane.
23. The heat of fusion of ice at its normal melting point is 3.3 x
10 5 joules/kg and the specific volume of ice is greater than the spe
cific volume of water at this point by 9 x 10~ 5 m 3 /kg. The value of
(l/V)(3V/3T)p for ice is 16 x 10" 5 per degree and its value of (1/V)
x (8V/9P) T is 12 x 10 11 (m 2 /newton).
(a) Ice at 2°C and at atmospheric pressure is compressed iso
thermally. Find the pressure at which the ice starts to melt.
(b) Ice at 2°C and atmospheric pressure is kept in a container at
constant volume and the temperature is gradually increased. Find the
temperature at which the ice begins to melt.
24. Considering that all constituents of a chemical reaction are per
fect gases obeying Eqs. (821), write out the expressions for In K, the
logarithm of the equilibrium constant, in terms of T, T , of the con
tents of integration g^Q and s^q per mole and of the v{s. Show that
the derivative of In K^ with respect to T at constant P is equal to
AH/RT 2 , where AH is the change in enthalpy (i.e., the heat evolved)
when v\ moles of substance M^ disappears in the reaction (a negative
value of i>i means the substance appears, i.e., is a product). This re
lation between [dln(K)/dT]p and AH is known as the van't Hoff equa
tion.
25. The probability that a certain trial (throw of a die or drop of a
bomb, for example) is a success is p for every trial. Show that the
probability that m successes are achieved in n trials is
P m (n) = — 77 — : — 77 p (1 p) (this is the binomial
m m!(nm)! ,. . ., ,. v
distribution)
Find the average number in of successes in n trials, the meansquare
(m 2 ) and the standard deviation Am of successes in n trials.
26. The probability of finding n photons of a given frequency in an
enclosure that is in thermal equilibrium with its walls is P n =
(1  a)a n f where a(0 <a < 1) is a function of temperature, volume of
the enclosure, and the frequency of the photons. What is the mean
number fi of photons of this frequency? What is the fractional devia
tion An/h of this number from the mean? Express this fractional de
viation in terms of h, the mean number. For what limiting value of n
does the fractional deviation tend to zero?
PROBLEMS 257
27. A molecule in a gas collides from time to time with another
molecule. These collisions are at random in time, with an average
interval r, the mean free time. Show that, starting at time t = (not
an instant of collision) the probability that the molecule has not yet
had its next collision at time t is e~V T . What is the expected time
to this next collision? Show also that the probability that its previous
collision (the last one it had before time t = 0) was earlier than time
T is e~ T / T . What is the mean time of this previous collision? Does
this mean that the average time interval between collisions is 2t ?
Explain the paradox.
28. In interstellar space, the preponderant material is atomic hy
drogen, the mean density being about 1 hydrogen atom per cc. What
is the probability of finding no atom in a given cc ? Of finding 3 at
oms? How many H atoms cross into a given cc, through one of its
1cm 2 faces, per second, if the temperature is 1°K? If T is 1000°K?
29. A closed furnace F (Fig. P29) in an evacuated chamber con
tains sodium vapor heated to 1000 °K. What is the mean speed v of the
vapor atoms? At t = an aperture is opened in the wall of the furnace,
allowing a collimated stream of atoms to shoot out into the vacuum.
&
Fig. P29
The aperture is closed again at t = r. A distance L from the aperture,
a plate is moving with velocity u, perpendicular to the atom stream,
so that the stream deposits its sodium atoms along a line on the plate;
the position of the stream that strikes at time t hits the line at a point
a distance X = ut from its beginning. Obtain a formula for the density
of deposition of sodium as function of X along the line assuming that
r <C (L/v), and find the value of X for which this density is maximum.
Sketch a curve of the density versus X.
30. Most conduction electrons in a metal are kept from leaving
the metal by a sudden rise in electric potential energy, at the surface
of the metal, of an amount eW , where W is the electric potential
difference between the inside and the outside of the metal. Show that
if the conduction electrons inside the metal are assumed to have a
Maxwell distribution of velocity, there will be a thermionic emission
current of electrons from the surface of a metal at temperature T
that is proportional to /T exp (eW /kT). What is the velocity distri
bution of these electrons just outside the surface ? [The measured
258
THERMAL PHYSICS
thermionic current is proportional to T 2 exp (e^/kT), where
<p< W ; see Problem 63.]
0)
c
©
o
a
eW f
exterior
metal surface
interior
Fig. P30
31. A gas of molecules with a Maxwell distribution of velocity at
temperature T is in a container having a piston of area A, which is
moving outward with a velocity u (small compared to v), expanding
the gas adiabatically (Fig. P31). Show that, because of the motion
of the piston, each molecule that strikes the piston with velocity v at
Fig. P31
an angle of incidence 6 rebounds with a loss of kinetic energy of an
amount 2mvu cos 6 (u <v). Show that consequently the gas loses en
ergy, per second, by an amount dU = PAu = P dV, where dV is the
increase in volume of the container per second.
32. Helium atoms have a collision cross section approximately
equal to 2 x 10~ 16 cm 2 . In helium gas at standard conditions (1 atm
pressure, 0°C), assuming a Maxwell distribution, what is the mean
speed of the atoms? What is their mean distance apart? What is the
mean free path? The mean free time?
33. A gas is confined between two parallel plates, one moving with
respect to the other, so that there is a flow shear in the gas, the mean
gas velocity a distance y from the stationary plate being /3y in the x
direction (Fig. P33). Show that the zeroorder velocity distribution
in the gas is
PROBLEMS 259
^^^^^ ^^ v =/3L
I I
v = £y
^^^^^^^^^^^ ^^ v =
x — ^
Fig. P33
(l/277mkT) 3/2 exp{(l/2mkT)[(p x  m/3y) 2 + p 2 , + p]} =
= f (p x m/3y, p y , p z )
Use Eq. (143) to compute f to the first order of approximation. Show
that the mean rate of transport of x momentum across a unit area
perpendicular to the y axis is (n/V)(t c /3/m)J ff p 2 f dp x dp y dp z
= (N/V)t c j3kT, which equals the viscous drag of the gas per unit area
of the plate, which equals the gas viscosity 77 times /3, the velocity
gradient. Express 77 in terms of T and A (mean free path) and show
that the diffusion constant D of Eq. (148) is equal to (77/p), where p
is thejiensity of the gas.
34. Use the Maxwell Bo Itzmann distribution to show that, if the
atmosphere is at uniform temperature, the density p and pressure P
a distance z above the ground is exp(mgz/kT) times p and P , re
spectively (where g is the acceleration of gravity). Express p and
P in terms of g, T, and M a , the total mass of gas above a unit ground
area. Obtain this same expression from the perfect gas law, P = pkT/m
and the equation dP = pg dz giving the fall off of pressure with height
(assuming T is constant). Find the corresponding expressions for p
and P in terms of z if the temperature varies with pressure in the
way an adiabatic expansion does, i.e., P = (p/C)>, T = (Dp)>~ *, where
y =(C p /C v )[seeEqs. (412)].
35. A tube of length L = 2 m and of cross section A = 10" 4 m 2 con
tains C0 2 at normal conditions of pressure and temperature (under
these conditions the diffusion constant D for C0 2 is about 10 m 2 /sec).
Half the C0 2 contains radioactive carbon, initially at full concentra
tion at the lefthand end, zero concentration at the righthand end, the
concentration varying linearly in between. What is the value of t c for
C0 2 under these conditions ? Initially, how many radioactive mole
cules per second cross the midpoint cross section from left to right?
[Use Eqs. (147).] How many cross from right to left? Compute the
difference and show that it checks with the net flow, calculated from
the diffusion equation (net flow) = D(dn/dx).
36. The collision cross section of an air molecule for an electron
is about 10" 19 m 2 . At what pressure will 90 per cent of the electrons
emitted from a cathode reach an anode 20 cm away?
260 THERMAL PHYSICS
37. A gasfilled tube is whirled about one end with an angular ve
locity co. Find the expression for the equilibrium density of the gas
as a function of the distance r from the end of the tube.
38. A vessel containing air at standard conditions is radiated with
xrays, so that 0.01 per cent of its molecules are ionized. A uniform
electric field of 10 4 volts/meter is applied. What is the initial net flux
of electrons ? Of ions? (See Problem 36 for the cross section for
electrons; the cross section for the ions is four times this. Why?)
What is the ratio between drift velocity and mean thermal velocity for
the electrons ? For the ions ?
39. A solid cylinder of mass M is suspended from its center by a
fine elastic fiber so that its axis is vertical. A rotation of the cylinder
through an angle 6 from equilibrium requires a torque KB to twist
the fiber. When suspended in a gasfilled container at temperature T,
the cylinder exhibits rotational fluctuation due to Brownian motion.
What is the standard deviation (A6) of the amplitude of rotation and
what is the standard deviation (Aco) of its angular velocity? What
would these values be if the container were evacuated?
40. Observations of the Brownian motion of a spherical particle of
radius 4 x 10~ 7 m in water, at T = 300°K, and of viscosity 10~ 3 newton
sec/m 2 were made every 2 sec. The displacements in the x direction,
x(t)  x(t 2), were recorded and were tabulated as shown in Table
P40.
TABLE P40
<5 = x(t)  x(t
 2)
No.
times this
in units of 10"
• 6 m
was
» observed
Less than ±0.5
111
Between 0.5 and 1.5
87
0.5
1.5
95
1.5
2.5
47
1.5
2.5
32
2.5
3.5
8
2.5
3.5
15
3.5
4.5
3
3.5
4.5
2
4.5
5.5
4.5
5.5
1
Larger than ±
5.5
Compute the mean value of 6 and its standard deviation. How close is
this distribution to a normal distribution [Eq. (1117)] ? Use Eq.
(1511) to compute Avogadro's number from the data, assuming
R = kN is known.
PROBLEMS 261
41. Show that if the Hamiltonian energy of a molecule depends on a
generalized coordinate q or momentum p in such a way that H — °°
as p or q — ±°°, it is possible to generalize the theorem on equipar
tition of energy to
If) 41?) '«
M 'av 'av
Verify that this reduces to ordinary equipartition when H has a quad
ratic dependence on q or p. If H has the relativistic dependence on
the momentum
H = c V(Px + Py + Pz) + m2c2
show that
(c a p£/H) av =  = (c 2 p/H) 2 av = kT
42. A harmonic oscillator has a Hamiltonian energy H related to
its momentum p and displacement q by the equation
p 2 + (mcuq) 2 = 2mH
When H = E, a constant energy, sketch the path of the system point in
twodimensional phase space. What volume of phase space does it en
close? In the case of N similar harmonic oscillators, which have the
total energy E given by
N N
Vp]+V (Mu>qi) 2 = 2ME
j=l ' j=l
with additional coupling terms, too small to be included, but large
enough to ensure equipartition of energy, what is the nature of the path
traversed by the system point ? Show that the volume of phase space
"enclosed" by this path is (l/N!)(27rE/a)) N .
43. A gas of N point particles, with negligible (but not zero) colli
sion interactions, enclosed in a container of volume V, has a total en
ergy U. Show that the system point for the gas may be anywhere on a
surface in phase space which encloses a volume V (277 mU) /
(3N/2) !. For an ensemble of these systems to represent an equilibrium
state, how must the system points of the ensemble be distributed over
this surface ?
44. A system consists of three distinguishable molecules at rest,
each of which has a quantized magnetic moment, which can have its z
component +M, 0, or M. Show that there are 27 different possible
262 THERMAL PHYSICS
states of the system; list them all, giving the total z component M z i
of the magnetic moment for each. Compute the entropy S = kEf j
x ln(fi) of the system for the following a priori probabilities fj:
(a) All 27 states equally likely (no knowledge of the state of the
system).
(b) Each state is equally likely for which the z component M z of
the total magnetic moment is zero; f i = for other states (we know
that M z = 0).
(c) Each state is equally likely for which M z = M; fj_ = for all
other states (we know that M z = M).
(d) Each state is equally likely for which M z = 3M; f j = for all
other states (we know that M z = 3M).
(e) The distribution for which S is maximum, subject to the re
quirements that Efi = 1 and the mean component EfiM z i is equal to
yM. Show that for this distribution
f i = exp [(3M  M zi ) of/(l + x + x 2 ) 3 ]
where x = e aM (a being the Lagrange multiplier) and where the value
of x (thus of a) is determined by the equation y = 3(x 2  l)/(l+x + x 2 ).
Compute x and S for y  3, y = 1, and y = 0. Compare with a, b, c,
and d.
45. Suppose the atoms of the crystal of Eqs. (206) are sufficiently
"decoupled" so that it is a better approximation to consider the sys
tem as a collection of v  3N harmonic oscillators, all having the
same frequency co. Show that the partition function in this case is
Eo/kT//, ha;/kT\ 3N
Z = e "
where
yjieWkT)
E = [(V  V ) 2 /2kV ] + NBw
Compute the entropy, internal energy, and heat capacity of this system
and obtain approximate formulas for kT small and also kT large com
pared to "hcu. For what range of temperature do these formulas differ
most markedly from those of Eq. (2014)?
46. The atoms of the crystal of Problem 45 have spin magnetic mo
ments, so that the allowed energy of interaction with the magnetic field
of each atom is JCM z = ± (l/2)m5c, the magnets being parallel or anti
parallel respectively to the magnetic field 3C. Show that for this system
the canonical ensemble yields the following expression for the Helm
no ltz function:
F = E + 3NkT in (l  e"^ /kT )  km*  NkT ln(l + e  mCK / kT )
PROBLEMS 263
— E + 3NkT ln(ftwAT)  NkT In 2  (Nm3C 2 /8kT)
kT ^> Ho) and m3C
Compute S, C v , and U for the high temperature limit and compare
the results with Problem 19. What is the magnetization snx for this
system ? What is the rate of change of T with respect to OC for adia
batic demagnetization of this crystal at constant volume?
47. Use the final result of Problem 42 to show that the entropy of
N distinguishable harmonic oscillators, according to the microcanon
ical ensemble (every system in the ensemble has energy NkT) is
S = Nk[l + ln (kT/hw)]
48. For the solid described by Eq. (2016) show that P =[(V  V)/
*V ] + (>U D /V), where U = [(V  V) 2 /2/cV ] + U D and y = (V/6) x
(d#/dV). Thence show that, for any temperature, if y is independent
of temperature, the thermal expansion coefficient (3 is related to y
by the formula
13 = (l/V)0V/3T) p = kOP/3T) v = (KyCy/V)
Constant y is called the Gruneisen constant.
49. A system consists of a box of volume V and a variable num
ber of indistinguishable (MB) particles each of mass m. Each particle
can be "created" by the expenditure of energy y; once created it be
comes a member of a perfect gas of point particles within the volume
V. The allowed energies of the system are therefore ny plus the ki
netic energies of n particles inside V, for n = 0, 1, 2, .... Show that
the Helmholtz function for this system (canonical ensemble) is
F = kT In
£ (v^ n /nl)
n =
■kTVX
where X = (27rmkT/h 2 ) 3/2 e y ' . Calculate the probability that ^par
ticles are present in the box and thence obtain an expression for N,
the mean number of free particles present as a function of y, T, and
V. Also calculate S, C v , and P from F and express these quantities
as functions of N, T, and V.
50. What fraction of the molecules of H 2 gas are in the first excited
rotational state (i= 1) at20°K, at 100°K, and at 5000°K? What are
the corresponding fractions for 2 gas ? What fraction of the mole
cules of H 2 gas are in the first excited vibrational states (n = 1) at
20 °K and 5000 °K? What are the corresponding fractions for Q 2 gas ?
264 THERMAL PHYSICS
51. Plot the heat capacity C v , in units of Nk for 2 gas, from 100
to 5000 °K.
52. The solid of Eqs. (2014) sublimes at low pressure, at a subli
mation temperature T s which is large compared to 6, the resulting
vapor being a perfect diatomic gas, with properties given by Eqs.
(2114) and (222) (where rot < T s <0 v ib). Show that the equation
relating T s and the sublimation pressure P s is approximately
G S * V P S + f Nktf + 3NkT s ln(0/T s )  NkT s
= G g  NkT s ln(P s V o T s 7/2 /N o k0 5 / 2 )  N kT s
where the equation
N = V (47rIeke/h 2 )(27Tmke/h 2 ) 3/2
defines the constant N . Since V « V g = NkT s /P s and 6 <C T s , show
that this reduces to
p s « N k Vre/v
Also show that the latent heat of sublimation
L s = T s (S g  S s )  NkT s
53. Work out the grand canonical ensemble for a gas of point atoms,
each with spin magnetic moment, which can have magnetic energy
+ (1/2) /u JC or (1/2) /iJC in a magnetic field 3C in addition to its kinetic
energy. Obtain the expression for N and expressions for Q , jul, S, U,
C v , and the equation of state, in terms of N, T, and 5C. How much
heat is given off by the gas when the magnetic field is reduced from 3C
to zero isothermally, at constant volume ?
54. A system consists of three particles, each of which has three
possible quantum states, with energy 0, 2E, or 5E, respectively.
Write out the complete expression for the partition function Z for
this system: (a) if the particles are distinguishable; (b) if the parti
cles obey Maxwell Bo ltzmann statistics; (c) if they obey Einstein
Bose statistics; (d) if they obey Fermi Dirac statistics. Calculate the
entropy of the system in each of these cases.
55. The maximum intensity per unit frequency interval, in the sun's
spectrum, occurs at a wavelength of 5000 A. What is the surface tem
perature of the sun?
56. Show that, for EinsteinBose particles (bosons)
PROBLEMS 265
S=k? [n± In (hi)  (1 + hi) In (1 + hi)]
i
57. It has been reported that the fission bomb produces a tempera
ture of a million °K. Assuming this to be true over a sphere 10 cm in
diameter: (a) What is the radiantenergy density inside the sphere?
(b) What is the rate of radiation from the surface? (c) What is the
radiant flux density 1 km away? (d) What is the wavelength of maxi
mum energy per unit frequency interval ?
58. The Planck distribution can be obtained by considering each
standing electromagnetic wave in a rectangular enclosure (L x L y L z )
as a degree of freedom, with coordinate Q„ proportional to the ampli
tude of the electric vector, with momentum V v proportional to the
amplitude of the magnetic vector and with a field energy, correspond
ing to a term in the Hamiltonian, equal to 27rc 2 Pf,+ (cof;/87rc 2 )Q^, where
c is the velocity of light and where the allowed frequency of the vth
standing wave is given by
wj, = tt 2 c 2 [(VL x ) 2 + (m^/Ly) 2 + (n„/L z ) 2 ]
(because of polarization, there are two different waves for each trio
ky, m^, n^). Use the methods of Eqs. (204) to (2011) to prove that
the average energy contained in those standing waves with frequencies
between a> and a> + dco is dE = (H/7T 2 c 3 ) a) 3 da>/(e' n ^/ kT  1). Compare
this derivation with the one dealing with photons, which produced Eq.
(253).
59. Analyze the thermal oscillations of electromagnetic waves
along a conducting wire of length L. In this case of one dimensional,
standing waves, the nth wave will have the form cos (7rnx/L)e iu)t ,
where cu = 2iif = 7rnc/L, c being the wave velocity and f the frequency
of the nth standing wave. Use a one dimensional analogue of the
derivation of Problem 58 to show that the energy content of the waves
with frequencies between f and f + df is [2Lhf df/c(e M A T  1)] . If
the wire is part of a transmission line, which is terminated by its
characteristic impedance, all this energy will be delivered to this im
pedance in a time 2L/C. Show, consequently, that the power delivered
to the terminal impedance, the thermal noise power, in the frequency
band df at frequency f is hf df/(e nf / kT  1). Show that this reduces
to the familiar uniform distribution (white noise) at low frequencies,
but that at high frequencies the power drops off exponentially. Below
what frequency is the noise power substantially "white" at room tem
peratures (300 °K)?
60. A container of volume V has N shortrange attractive centers
(potential wells) fixed in position within the container. There are also
bosons within the container. Each particle can either be bound to an
266 THERMAL PHYSICS
attractive center, with an energy y (one level per center), or can be
a free boson, with energy equal to its kinetic energy, E. Use the anal
ysis of Chapter 30 to show that the equation relating the mean number
N of bosons to their chemical potential /i is
N = Q (y  M )/kT : 1 + 1.129N f 1/2 (); N = g v(^)
Draw curves for ji/kT as a function of N /N for N/N = 1 and for
y/kT = 0.1 and 1.0, using Table 25.1. Draw the corresponding curves
for PV/NkT.
61. Suppose the particles of Problem 60 are MB particles instead
of bosons. Calculate the partition function Z for a canonical ensem
ble and compare it with the Z for Problem 49.
62. Show that, for Fermi Dirac particles (fermions),
S =  T [fli In (hi) + (1  hi) In (1  hi)]
63. The conduction electrons of Problem 30 are, of course, fer
mions. Show that, for FD statistics, the thermionic emission current
from the metal surface at temperature T is proportional to
T 2 exp(e0/kT), where 0= W  \x « w  jlx is called the thermi
onic work function of the surface.
64. The container and N attractive centers of Problem 60 have N
fermions, instead of bosons, in the system. By using Eqs. (267) and
(268) show that the equation relating \x and T and V is
N = ( _ ^ kT + N 7? (mAT), N as in Problem 60
Plot jx/kT as function of N /N for y/kT = 0.1 and 1.0, using Table
261. Draw the corresponding curves for PV/NkT.
65. Calculate the heat capacity of D 2 as a function of T/6 T0 ^ from
Otol.
66. The Schrodinger equation for a onedimensional harmonic os
cillator is
Its allowed energies and corresponding wave functions are
E n =J h(jj
(4)
PROBLEMS 267
i// n (x) = ]/moj/2 n nl^^ H n (xlt/R) exp (ma;x 2 /2Ti)
where H (z) = 1, H^z) = 2z, H 2 (z) = 4z 2  2, ^(z) = 8z 3  12z, etc.
Two identical, onedimensional oscillators thus have a Schrodinger
equation
H(x,y)* =  £ (£ 2 + $) * + ± m u,*(x* + y*)* = E*
where x is the displacement of the first particle from equilibrium
and y that of the second.
(a) Show that allowed solutions of this equation, for the energy
liu>(n + 1/2), may be written either as linear combinations of the prod
ucts ip m (x)i// n _ m (y) for different values of m between and n, or
else as linear combinations of the products
(b) Express the solutions
and
for m = 0, 1, and 2 as linear combinations of the solutions \p m (x)\p n (y)
for m,n = 0,1,2.
(c) Which of these solutions are appropriate if the two particles
are bosons ? Which if they are fermions ?
(d) Suppose the potential energy has an interparticle repulsive
term (l/2)m/t 2 (x  y) 2 (where k 2 < a; 2 ) in addition to the term
(l/2)mcD 2 (x 2 + y 2 ). Show that, in this case, the allowed energies for
bosons differ from those for fermions. Which lie higher and why?
Constants
Gas constant R = 1.988 kgcal/moledeg.
= 8.314 x 10 3 joules/moledeg.
Avogadro's number N (No. per kg mole) = 6.025 x 10 26
Number of molecules per m 3 at standard conditions = 2.689 x 10 25
Standard conditions: T = 0°C = 273.16°K, P = 1 atm
Volume of perfect gas at standard conditions = 22.41 m 3 /mole
Dielectric constant of vacuum € = (1/9) x 10 9 farads /m
Electronic charge e = 1.602 x 10" 19 coulomb
Electron mass m = 9.106 x 10" 31 kg
Proton mass = 1.672 x 10" 27 kg
Planck's constant h = 6.624 x 10" 34 joulesec
fi= (h/27i) = 1.054 x lO" 34
Boltzmann's constant k= 1.380 x 10~ 23 joule/°K
Acceleration of gravity g = 9.8 m/sec 2
1 atm = 1.013 x 10 5 newtons/m 2
1 cm Hg = 1333 newtons/m 2
1 new ton = 10 5 dynes
1 joule = 10 7 ergs
1 electronvolt (ev) = 1.59 x 10 19 joules
= k(7500°K)
Velocity of 1 ev
electron = 5.9 x 10 5 m/sec
1 kgcal = 4182 joules
Ratio of proton
mass to elec
tron ma ss = 1836
v = V8kT/7im = 1.96 x 10 4 m/sec for electrons at T = 1°K
= 146 m/sec for protons at T = 1°K
h 2 N 2/3 /27imk = 3961 °K m 2 for electrons, = 2.157°K m 2 for protons
"= 2.49xl0 5 m 3 for electrons
.= 3.17 m 3 for protons [seeEq. (2411)]
268
h 3 N /(27imkT) 3/2 = V b /T 3 / 2 ; V b
CONSTANTS 269
(Mo AT)(VN /N) 2 / 3 = (h72mkT)(3N /8tf) 2 /3 = A d /T
A d = 3017 m 2 for electrons [see Eq. (313)]
Glossary
Symbols used in several chapters are listed here with the numbers
of the pages on which they are defined.
a Van der Waals' constant, H
18, 193 H
a Stefan's constant, 55, 225
A area, 10 3C
b Van der Waals' constant, I
18, 193 J
(B magnetic induction, 15 k
c specific heat, 15 L
C heat capacity, 15 m
d perfect differential, 21 M
d imperfect differential, 22 am
D Curie constant, 20, 115 n
D diffusion constant, 121, 133 nj
D(x) Debye function, 178 N
e = 2.7183, nat. log. base
e charge on electron, 118 N
E energy of system, 145 p
8 potential difference, 83 pi
8 electric intensity, 118 P
f distribution function, 91, (P
112, 141 q
F Helmholtz' function, 62, Q
165 Q
JF Faraday constant, 83 r
g multiplicity of state, 167, r
227 R
G Gibbs' function, 63 S
h Planck's constant, 55 t
•h = h/2?r t c
hi scale factor, 108 T
enthalpy, 52, 60
Hamiltonian function, 109,
141, 239
magnetic intensity, 15
amount of information, 195
torsion in bar, 15
Boltzman constant, 17, 147
latent heat, 70
mass of particle, 10
mobility, 119
magnetization, 15
number of moles, 14
occupation number, 208
number of particles, 10,
209
Avogadro's number, 13
momentum, 96, 109, 141
partial pressure, 80
pressure, 9, 165
magnetic polarization, 15
coordinate, 108, 141
heat, 8
collision function, 104, 116
internuclear distance, 190
position vector, 103
gas constant, 16
entropy, 40, 45, 147, 165
time, 104
relaxation time, 116
temperature, 7, 17, 38
270
GLOSSARY
271
U internal energy, 9, 25, 59
U drift velocity, 119
v velocity, 10
V volume, 10
W work, 23
x,y,z coordinates, 103
Z normalizing constant, 110
Z partition function, 165, 182
3 grand partition function,
204
a Lagrange multiplier, 151,
164
jS thermal expansion coef
ficient, 19
y C p /C v , 19
€ particle energy, 181, 208
7] heat efficiency, 34
77 viscosity, 131
6 Debye temperature, 178
k compressibility, 19
A mean free path, 102
li chemical potential, 16, 45,
79, 205
li magnetic moment, 113
li permeability of vacuum, 15
77
P
a
o
7
*(q)
X
W J
b
In
n!
number of degrees of free
dom, 108
quantum number, 145, 208
stoichiometric coefficient,
78
= 3.1416
density, 128, 235
standard deviation, 90
collision cross section,
101
mean free time, 102
potential energy, 105
number of degrees of
freedom, 141
magnetic susceptibility, 15
concentration, 80
wave function, 238
277 (frequency), 55, 170
oscillator constant, 124,
175
grand potential, 63, 204
approximately equal to
average value, 89
partial derivative, 20
natural logarithm, 44
factorial function, 94, 156,
159
Index
Absolute zero, 17
Adiabatic process, 29, 34, 45,
230, 251, 259
Antisymmetric wave functions,
241, 267
Average energy, 110, 164, 203
kinetic energy, 10, 123
value, 89
Avogadro's number, 13, 260,268
Binomial distribution, 89, 256
Black body radiation, 54, 221,
265
Boltzmann constant, 17, 147, 268
equation, 104, 116
Maxwell (see Maxwell Boltz
mann distribution)
BoseEinstein statistics, 212,
264
boson, 212, 220, 241, 266
Calculations, thermodynamic, 64
Canonical ensemble, 163, 202, 262
Capacity, heat, 14, 113, 172, 197,
201, 231
at constant magnetization, 29,
56, 255
at constant pressure, 15, 28,
65, 251
at constant volume, 15, 28, 65,
113, 236, 247, 255
Carnot cycle, 33, 40, 252
Centigrade temperature scale,
7, 16
Chemical potential, 16, 45, 56,
67, 79, 203, 211, 234
Chemical reactions, 78
Clausius' principle, 37
ClausiusClapeyron equation,
71, 256
Collision, 97, 101
Collision function, 104, 116
Compressibility, 19, 112, 155, 179
Concentration, 80
Condensation, Einstein, 228, 232
Conditional probability, 68, 131
Conductivity, electric, 118, 260
Continuity, equation of, 104, 142
Critical point, 74
Cross section, collision, 101,259
Crystal, 111, 154, 170, 262
Curie's law, 20, 56, 113
Cycle
Carnot, 33, 40
reversible, 34
Debye formulas for C v , 177,234
Debye function, 178
Degenerate state, 216, 230, 234
Degrees centigrade, 7, 16
Degrees of freedom, 108, 222
Degrees Kelvin, 17, 38
Density
fluctuations, 127, 206
272
INDEX
273
Density, probability, 91
Dependent variables, 6
Derivative, partial, 20, 65
Deuterium, 248
Deviation, standard, 90, 98
Diatomic molecule, 195
Differential
imperfect, 22
perfect, 21
Diffusion, 120, 133, 259
Dirac (see FermiDirac statis
tics)
Distinguishable particles, 186,
210
Distribution function, 89, 112,
141, 147, 216
binomial, 89, 256
geometric, 218, 256
Maxwell, 98, 160, 258
Maxwell Boltzmann, 103, 105,
217, 233, 259
momentum, 94
normal, 93, 98
Planck, 222, 265
Poisson, 91, 206, 218
Drift velocity, 98, 119, 260
Efficiency of cycle, 34
Einstein (see BoseEinstein sta
tistics)
condensation, 231
formula for C v , 171
Electric conduction in gases,
118, 260
Electrochemical processes, 82
Electromagnetic waves, 54, 221,
265
Electrons, 213, 234, 244, 266
Energy, 9
density, 54, 222
distribution in, 103, 165, 217,
233
Hamiltonian, 109, 141, 239,261
internal, 9, 12, 23, 59, 155,
164, 204, 251
kinetic, 10, 108, 183
magnetic, 16, 125
Energy, potential, 105, 123
rotational, 195
vibrational, 112, 155, 200
Engine, heat, 33
Ensemble, 141, 147, 154, 163
canonical, 163, 202, 262
grand canonical, 202, 264
microcanonical, 154, 261
Enthalpy, 52, 60
Entropy, 40, 69, 76, 147, 165,203
of mixing, 49, 254
of perfect gas, 45
of universe, 44, 255
Equation of state, 16, 18, 64, 165,
179, 193, 204, 250
Equilibrium constant, 81
state, 6, 59
Equipartition of energy, 110, 122
Euler's equation, 42, 64
Evaporation, 72
Expansion coefficient, thermal,
19, 250
Expected value, 89
Extensive variables, 14, 41, 59
Factorial function, 94, 156, 159
FermiDirac statistics, 213, 233,
264
Fermion, 212, 233, 242, 266
Fluctuations, 122, 206, 216, 226,
260, 265
Fokker Planck equation, 133
Gas
boson, 220
constant, R, 16, 268
fermion, 233
fluctuations in, 127, 206
paramagnetic, 56
perfect, 10, 17, 30, 45, 51, 157,
205, 252
photon, 54, 221, 265
statistical mechanics of, 181
Van der Waals, 18, 48, 193,
251, 255
Geometric distribution, 218, 256
Gibbs' function, 63, 70, 77, 203
274
INDEX
GibbsHelmholtz equation, 83
Grand canonical ensemble, 202,
264
Grand partition function, 203,
209
Grand potential, 63, 204, 211,
213
Griineisen constant, 263
Hamiltonian function, 109, 141,
239, 261
Hamilton's equations, 142, 173
Heat, 8, 24
capacity, 14, 28, 56, 65, 113,
231, 236, 247
engine, 33
reservoir, 26, 29, 34
specific, 15, 172, 197, 201,229
Heisenberg principle, 145
Helium, 212, 232, 244
Helmholtz function, 62, 165, 178,
189, 204
Hydrogen gas, 196, 200, 245
Hyper sphere, volume of, 159
Independent variables, 6
Indistinguishability, 185
Information, 147
Integrating factor, 22, 40
Intensive variables, 14, 41
Interaction between particles,
190
Internal energy, 9, 12, 23, 59,
155, 164, 204, 251
Inversion point, 52
Irreversible process, 42
Isothermal process, 29, 33
Joule coefficient, 47
experiment, 43, 46
Joule Thomson coefficient, 51
experiment, 51
Kelvin, degrees, 17, 38
Kelvin's principle, 35
Kinetic energy, 10, 108, 183
Kinetic energy, per degree of
freedom, 110, 122
Lagrange multipliers, 150, 164,
203
Langevin equation, 131
Latent heat, 70
Law of mass action, 81
Law of thermodynamics
first, 25
second, 32, 38
Legendre transformation, 61
Liouville's theorem, 131
Macrostate, 141
Magnetic induction, 15
Magnetic intensity, 15, 20
Magnetization, 15, 20, 30, 113,
263
Mass, 10
Mass action, law of, 81
Maxwell distribution, 90, 160,
258
Maxwell/s relations, 60, 62
Maxwell Boltzmann
distribution, 102, 165, 217,
233, 259, 264
particles, 209, 211, 217, 231
Mean free path, 102
time, 102, 257
Melting, 69
latent heat of, 70
Metals, 180, 234, 244, 266
Metastable equilibrium, 243
Microcanonical ensemble, 154,
261
Microstate, 5, 141, 146
Mixing, entropy of, 49, 254
Mobility, 119, 121
Moment, magnetic, 113, 262
Momentum, 11, 55, 142, 173
distribution (see Maxwell
Boltzmann distribution)
of photon, 55
space, 96
Multiplicity, 167, 185, 227, 233
INDEX
275
Natural variables, 41, 57
Newton, 9, 268
Noise, thermal, 127, 206, 265
Normal distribution, 93, 98
Normal modes of crystal, 175
Normalizing constant, 110, 238
Numerator, bringing to, 65
Occupation numbers, 208, 218,
220
Ohm's law, 119
Omega space, 176
Or tho hydrogen, 245
Orthogonal functions , 238
Oscillator, simple, 111, 154, 170,
261
fluctuations of, 125
Oxygen gas, 102, 196, 200
Parahydrogen, 245
Paramagnetic substance, 16, 20,
30, 56, 113, 251, 255
Partial derivative, 20, 65
Particle states, 185, 208
Partition function, 165, 182, 203,
209
Path, mean free, 102
Pauli principle, 213, 218, 242
Perfect differential, 21
Perfect gas, 17, 51, 96, 152, 157,
205
of point particles, 10, 30, 45
Phase, change of, 68
Phase space, 103, 141, 144
Phonons, 174
Photon gas, 54, 212, 221, 265
Planck distribution, 222, 265
Planck's constant, 55, 222, 268
Poisson distribution, 91, 206,
218
Polarization, magnetic, 15, 114
Potential
chemical, 16, 45, 56, 67, 79,
203, 211, 234
energy, 105, 123, 191, 199
grand, 63, 204, 211, 213
Potential, thermodynamic, 59,
64
Pressure, 9, 55, 75, 106, 179,
193, 230, 235
atmospheric, 259, 268
partial, 73, 80
radiation, 55, 225
vapor, 73
Probability, 87, 149
Process
adiabatic, 29, 34, 45, 230, 251,
259
electrochemical, 82
irreversible, 43
isothermal, 29, 33
quasistatic, 24, 27
reversible, 34, 40
spontaneous, 42
Protons, 213, 243
Quantum states, 144, 167
statistics, 208, 239
Quasistatic process, 24, 27
Radiation, thermal, 54, 221, 265
fluctuation in, 226, 265
Random walk, 90, 129
Rayleigh Jeans distribution, 223
Reactions, chemical, 78
Relaxation time, 116
Reservoir
heat, 26, 29, 34
work, 27
Reversible process, 34, 40
Rotational partition function, 195
Sakur Tetrode formula, 189
Scale
centigrade, 7, 16
factor, 108
Kelvin, 17, 38
thermodynamic, 38, 40
Schrodinger's equation, 144, 183,
238, 266
Solid state, 19, 68, 111, 154, 170,
266
276
INDEX
Space
momentum, 96
phase, 103, 141, 144
Specific heat, 15, 172, 197, 201,
229, 236, 247
Spin, 227, 233
Spontaneous process, 42
Standard deviation, 90, 98
State
equation of, 16, 18, 64, 165,
179, 193, 204, 250
macro, 141
micro, 5, 141, 146
system, 185, 208
variables, 6, 13
Stefan's law, 55, 225
Stirling's formula, 95, 156
Stokes' law, 131
Stoichiometric coefficient, 78
Sublimation, 74, 264
Susceptibility, magnetic, 15
Symmetric wave functions, 240,
267
System state, 185, 208
Taylor's series, 105
Temperature, 7
Debye, 178
of melting, 70
scale, 7, 17, 38, 40
thermodynamic, 38, 40
of vaporization, 72
Time, mean free, 102, 257
Thermionic emission, 257, 266
Translational partition function,
183
Transport phenomena, 116
Triple point, 74, 256
Ultraviolet catastrophe, 223
Uncertainty principle, 145
Universe, entropy of, 44, 255
Van der Waals' equation, 18, 48,
198, 251, 255
Van't Hoff 's equation, 82, 256
Vapor pressure, 73
Vaporization, latent heat of, 72
Variables
dependent, 6
extensive, 14, 41, 59
independent, 6
intensive, 14, 41
state, 6, 13
Variance, 90, 99
Velocity distribution, 96, 160,
257
drift, 98, 119, 260
mean, 100
mean square, 123
Vibrational partition function, 199
specific heat, 179, 200
Vibrations
of crystal, 112, 155, 175, 200
of molecule, 199
Virial equation, 18
Viscosity, 131, 259
Walk, random, 90, 129
Wave functions, 183, 238
Weight, statistical, 192
Work, 23
Work reservoir, 27
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